Recognizing the flat torus among RCD*(0,N) spaces via the study of the first cohomology group - Nicola Gigli; Chiara Rigoni (Calculus of Variations and Partial Differential Equations, 2018)
We prove that if the dimension of the first cohomology group of a $\mathsf{RCD}^\star (0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings - Huynh, D. B. P.; Federico Pichi; Gianluigi Rozza (Numerical Methods for PDEs, 2018)

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings - D.B.P. Huynh; Federico Pichi; Gianluigi Rozza (SEMA SIMAI Springer Series, 2018)
In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; an a posteriori error estimation procedures—rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities—to model the materials and loads—and geometrical parameters—to model different geometrical configurations—with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation - Roberto Feola; Filippo Giuliani; Michela Procesi (2018)

Regularity estimates for scalar conservation laws in one space dimension - Elio Marconi (Journal of Hyperbolic Differential Equations, 2018)
We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set w : f″(w)≠0 is dense, the regularity of the solution can be expressed in terms of BVΦ spaces, where Φ depends on the nonlinearity of f. If moreover the set w : f″(w) = 0 is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f′∘ u(t) ∈BV loc(ℝ) for every t > 0 and that this can be improved to SBVloc(ℝ) regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

Renormalization analysis for degenerate ground states - Hasler, David; Markus Lange (J. Funct. Anal., 2018)

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework - Andrea Cangiani; E.H. Georgoulis; Morozov, A. Yu.; Sutton, O. J. (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018)
Understanding how patterns and travelling waves form in chemical and biological reaction–diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction–diffusion models, such as Lotka–Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction–diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.

Second order differentiation formula on RCD(K, N) spaces - Nicola Gigli; Luca Tamanini (Rendiconti Lincei-Matematica e Applicazioni, 2018)

Second order differentiation formula on RCD*(K,N) spaces - Nicola Gigli; Luca Tamanini (2018)

Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition - Nicola Demo; Marco Tezzele; Gianluca Gustin; Gianpiero Lavini; Gianluigi Rozza (Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018)
Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method. We use the free form deformation (FFD) for an automated perturbation of the shape, and the finite volume method to simulate the multiphase incompressible flow around the deformed hulls. Further computational reduction is done by the dynamic mode decomposition (DMD) technique: from few high dimensional snapshots, the system evolution is reconstructed and the final state of the simulation is faithfully approximated. Finally the global optimization algorithm iterates over the reduced space: the approximated drag and lift coefficients are projected to the hull surface, hence the resistance is evaluated for the new hulls until the convergence to the optimal shape is achieved. We will present the results obtained applying the described procedure to a typical Fincantieri cruise ship.

Shape transitions in a soft incompressible sphere with residual stresses - Davide Riccobelli; Pasquale Ciarletta (Math. Mech. Solids, 2018)

On sinc quadrature approximations of fractional powers of regularly accretive operators - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (Journal of Numerical Mathematics, 2018)

Singular Hartree equation in fractional perturbed Sobolev spaces - Alessandro Michelangeli; Alessandro Olgiati; Raffaele Scandone (Journal of Nonlinear Mathematical Physics, 2018)
We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.

On some rigorous aspects of fragmented condensation - Daniele Dimonte; Marco Falconi; Alessandro Olgiati (2018)

Spectral triples on the Jiang-Su algebra - Jacopo Bassi; Ludwik Dabrowski (Journal of Mathematical Physics, 2018)

Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: The role of sample geometry - Noe Caruso; Aleksandar Cvetković; Alessandro Lucantonio; Giovanni Noselli; Antonio DeSimone (International Journal of Mechanical Sciences, 2018)
An elastic bilayer, consisting of an equibiaxially pre-stretched sheet bonded to a stress-free one, spontaneously morphs into curved shapes in the absence of external loads or constraints. Using experiments and numerical simulations, we explore the role of geometry for square and rectangular samples in determining the equilibrium shape of the system, for a fixed pre-stretch. We classify the observed shapes over a wide range of aspect ratios according to their curvatures and compare measured and computed values, which show good agreement. In particular, as the bilayer becomes thinner, a bifurcation of the principal curvatures occurs, which separates two scaling regimes for the energy of the system. We characterize the transition between these two regimes and show the peculiar features that distinguish square from rectangular samples. The results for our model bilayer system may help explaining morphing in more complex systems made of active materials.
Keywords: Bifurcation; Elastic bilayer; Pre-stretch; Shape programming

SRTP 2.0 - The evolution of the safe return to port concept - D. Cangelosi; A. Bonvicini; M. Nardo; Andrea Mola; A. Marchese; Marco Tezzele; Gianluigi Rozza (Technology and Science for the Ships of the Future - Proceedings of NAV 2018: 19th International Conference on Ship and Maritime Research, 2018)
In 2010 IMO (International Maritime Organisation) introduced new rules in SOLAS with the aim of intrinsically increase the safety of passenger ships. This requirement is achieved by providing safe areas for passengers and essential services for allowing ship to Safely Return to Port (SRtP). The entry into force of these rules has changed the way to design passenger ships. In this respect big effort in the research has been done by industry to address design issues related to the impact on failure analysis of the complex interactions among systems. Today the research activity is working to bring operational matters in the design stage. This change of research focus was necessary because human factor and the way to operate the ship itself after a casualty on board may have a big impact in the design of the ship/systems. Also the management of the passengers after a casualty is becoming a major topic for safety. This paper presents the state of the art of Italian knowledge in the field of system engineering applied to passenger ship address to safety improvement and design reliability. An overview of present tools and methodologies will be offered together with future focuses in the research activity.

Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs - D. Torlo; F. Ballarin; Gianluigi Rozza (SIAM-ASA Journal on Uncertainty Quantification, 2018)
In this work, we propose viable and eficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of the wRB (weighted reduced basis) method for stochastic parametrized problems with the stabilized RB (reduced basis) method, which is the integration of classical stabilization methods (streamline/upwind Petrov-Galerkin (SUPG) in our case) in the ofine-online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high-fdelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

Stochastic homogenisation of free-discontinuity problems - Filippo Cagnetti; Gianni Dal Maso; Lucia Scardia; Caterina Ida Zeppieri (2018)
In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems - Alexey Bolsinov; Lorenzo Guglielmi; Elena Kudryavtseva (Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018)
We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Time quasi-periodic gravity water waves in finite depth - Baldi, Pietro; Massimiliano Berti; Haus, Emanuele; Montalto, Riccardo (Invent. Math., 2018)

Transmission conditions obtained by homogenisation - Gianni Dal Maso; Giovanni Franzina; Davide Zucco (2018)
We study the asymptotic behaviour of solutions to variational problems in perforated domains with Neumann boundary conditions. We consider perforations that in the limit concentrate on a smooth manifold. We characterise the limits of the solutions and show that they solve a variational problem with a transmission condition across the manifold. This is expressed through a measure on the manifold, vanishing on sets of capacity zero. Then, we prove that every such measure can be obtained by homogenising suitable perforations. Eventually, we provide an asymptotic formula for this measure by using some auxiliary minimum problems.

Truncation and convergence issues for bounded linear inverse problems in Hilbert space - Noe Caruso; Alessandro Michelangeli; Paolo Novati (SISSA, 2018)
We present a general discussion of the main features and issues that (bounded) inverse linear problems in Hilbert space exhibit when the dimension of the space is infinite. This includes the set-up of a consistent notation for inverse problems that are genuinely infinite-dimensional, the analysis of the finite-dimensional truncations, a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

Two examples of minimal Cheeger sets in the plane - Leonardi, G. P.; Saracco, G. (Ann. Mat. Pura Appl. (4), 2018)

On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field - Paolo Bonicatto (Theory, Numerics and Applications of Hyperbolic Problems I, 2018)

Virtual element methods for elliptic problems on polygonal meshes - Andrea Cangiani; Sutton, Oliver J.; Gyrya, Vitaliy; Manzini, Gianmarco (Generalized barycentric coordinates in computer graphics and computational mechanics, 2018)

Weighted Cheeger sets are domains of isoperimetry - Saracco, G. (Manuscripta Math., 2018)

Wilson loop and its correlators in the limit of large coupling constant - Sysoeva, Ekaterina (Nucl. Phys. B, 2018)

Wilson loops and its correlators with chiral operators in $\mathcalN=2, 4$ SCFT at large $N$ - Sysoeva, Ekaterina (JHEP, 2018)

π-BEM : A flexible parallel implementation for adaptive , geometry aware , and high order boundary element methods - Nicola Giuliani; Andrea Mola; Luca Heltai (Advances in Engineering Software, 2018)

On the 1D wave equation in time-dependent domains and the problem of debond initiation - Giuliano Lazzaroni; Lorenzo Nardini (SISSA, 2017)
Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

Analytic geometry of semisimple coalescent Frobenius structures - Giordano Cotti; Davide Guzzetti (Random Matrices: Theory and Applications, 2017)
We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators - Guglielmo Feltrin; Fabio Zanolin (Topol. Methods Nonlinear Anal., 2017)

On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics - Giuseppe Pitton; Gianluigi Rozza (Journal of Scientific Computing, 2017)
In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

The approximation of parabolic equations involving fractional powers of elliptic operators - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (J. Comput. Appl. Math., 2017)

An avoiding cones condition for the Poincaré–Birkhoff Theorem - Alessandro Fonda; Paolo Gidoni (Journal of Differential Equations, 2017)
We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.
Keywords: Avoiding cones condition; Hamiltonian systems; Periodic solutions; Poincaré–Birkhoff theorem

Certi fied Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation - Denis Devaud; Gianluigi Rozza (Spectral and High Order Methods for Partial Differential Equations, 2017)
In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis  (IGA) is a growing research theme in scientic computing and computational mechanics, as well as reduced basis methods for parametric PDEs. Their combination enhances the solution of some class of problems, especially the ones characterized by parametrized geometries we introduced in this work. For a general overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate that it is also possible for some class of problems to deal with ane geometrical parametrization combined with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2]  sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work we show a certication of accuracy and a complete integration between IGA formulation and parametric certified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3 recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

On a certified smagorinsky reduced basis turbulence model - Rebollo, T.C.; E.D. Ávila; Marmol, M.G.; F. Ballarin; Gianluigi Rozza (SIAM Journal on Numerical Analysis, 2017)

The Cheeger constant of a Jordan domain without necks - Leonardi, G. P.; Neumayer, R.; Saracco, G. (Calc. Var. Partial Differential Equations, 2017)

Clifford Tori and the singularly perturbed Cahn–Hilliard equation - Matteo Rizzi (Journal of Differential Equations, 2017)
In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.
Keywords: Cahn–Hilliard equation; Clifford Torus; Lyapunov–Schmidt reduction; Willmore surface

Complex Friedrichs systems and applications - Nenad Antonić; Krešimir Burazin; Ivana Crnjac; Marko Erceg (2017)
We provide a suitable extension of the theory of abstract Friedrichs systems from real Hilbert spaces to the complex Hilbert space setting, which allows for applications to partial differential equations with complex coeffcients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous works, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Dirac-Klein-Gordon system, the Dirac-Maxwell system, and the time-harmonic Maxwell system, which are all addressed in the paper. Moreover, for the time-harmonic Maxwell system we also applied a suitable version of the two-field theory with partial coercivity assumption which is developed in the paper.

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology - Giuseppe Pitton; Annalisa Quaini; Gianluigi Rozza (Journal of Computational Physics, 2017)
<div class="abstract svAbstract " style="border: 0px; font-size: 16px; margin: 0px; padding: 0px; vertical-align: baseline; font-family: Arial, Helvetica, 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', STIXGeneral, 'Cambria Math', 'Arial Unicode MS', sans-serif; color: #2e2e2e;" data-etype="ab"> We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.
Keywords: Parametrized Navier–Stokes equations; Reduced basis method; Stability of flows; Symmetry breaking bifurcation

Computer simulations of phase field drops on super-hydrophobic surfaces - Livio Fedeli (Journal of Computational Physics, 2017)
We present a novel quasi-Newton continuation procedure that efficiently solves the system of nonlinear equations arising from the discretization of a phase field model for wetting phenomena. We perform a comparative numerical analysis that shows the improved speed of convergence gained with respect to other numerical schemes. Moreover, we discuss the conditions that, on a theoretical level, guarantee the convergence of this method. At each iterative step, a suitable continuation procedure develops and passes to the nonlinear solver an accurate initial guess. Discretization performs through cell-centered finite differences. The resulting system of equations is solved on a composite grid that uses dynamic mesh refinement and multi-grid techniques. The final code achieves three-dimensional, realistic computer experiments comparable to those produced in laboratory settings. This code offers not only new insights into the phenomenology of super-hydrophobicity, but also serves as a reliable predictive tool for the study of hydrophobic surfaces.
Keywords: Multigrid; Phase field; Quasi-Newton; Super-hydrophobicity

Conforming and nonconforming virtual element methods for elliptic problems - Andrea Cangiani; Manzini, Gianmarco; Sutton, Oliver J. (IMA J. Numer. Anal., 2017)

On contact interactions realised as Friedrichs systems - Marko Erceg; Alessandro Michelangeli (2017)
We realise the Hamiltonians of contact interactions in quantum mechanics within the framework of abstract Friedrichs systems. In particular, we show that the construction of the self-adjoint (or even only closed) operators of contact interaction supported at a fixed point can be associated with the construction of the bijective realisations of a suitable pair of abstract Friedrich operators. In this respect, the Hamiltonians of contact interaction provide novel examples of abstract Friedrich systems.

Coupling effects on the dynamic response of moored floating platforms for offshore wind energy plants - Giusti, Alessandro; Giovanni Stabile; Marino, Enzo; Claudio Borri (Procedia engineering, 2017)

Curvature terms in small time heat kernel expansion for a model class of hypoelliptic Hörmander operators - Davide Barilari; Elisa Paoli (Nonlinear Analysis, 2017)
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We completely describe the small time heat kernel expansions on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the drift field and the curvature-like invariants of the optimal control problem associated with the diffusion operator.
Keywords: Curvature; Hypoelliptic heat equation; Small time asymptotics

Curvature-adapted remeshing of CAD surfaces - Franco Dassi; Andrea Mola; Hang Si (Engineering with Computers, 2017)

The deal.II Library, Version 8.5 - Daniel Arndt; Wolfgang Bangerth; Denis Davydov; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Jean-Paul Pelteret; Bruno Turcksin; David Wells (JOURNAL OF NUMERICAL MATHEMATICS, 2017)

Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours - Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro (2017)
We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.

Discrete spectra for critical Dirac-Coulomb Hamiltonians - Matteo Gallone; Alessandro Michelangeli (2017)
The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.

Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3 - Felice Iandoli; Raffaele Scandone (Advances in Quantum Mechanics: Contemporary Trends and Open Problems, 2017)
The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q ≥ 3$.

On the effect of interactions beyond nearest neighbours on non-convex lattice systems - Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro (2017)
We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.

Effective Non-linear Dynamics of Binary Condensates and Open Problems - Alessandro Olgiati (Advances in Quantum Mechanics: Contemporary Trends and Open Problems, 2017)
We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

Elliptic diffeomorphisms of symplectic 4-manifolds - Vsevolod Shevchishin; Gleb Smirnov (2017)

Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening - Stefano Almi (ESAIM: Control, Optimisation and Calculus of Variations, 2017)
In the setting of planar linearized elasticity, we study a fracture model depending on the crack opening. Assuming that the crack path is known a priori and sufficiently smooth, we prove that the energy release rate is well defined. Then, we consider the problem of quasi-static evolution for our model. Thanks to a vanishing viscosity approach, we show the existence of such an evolution satisfying a weak Griffith’s criterion.

Friedrichs systems in a Hilbert space framework: solvability and multiplicity - Nenad Antonić; Marko Erceg; Alessandro Michelangeli (2017)
The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples.

Gamma-Convergence of Free-discontinuity problems - Filippo Cagnetti; Gianni Dal Maso; Lucia Scardia; Caterina Ida Zeppieri (SISSA, 2017)
We study the Gamma-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Gamma-convergence, representation of the Gamma-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants - Mikhail Bershtein; Giulio Bonelli; Massimiliano Ronzani; Alessandro Tanzini (Journal of Geometry and Physics, 2017)
We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between $\mathcal{N}=2$ supersymmetric gauge theories and two-dimensional conformal field theory. Talk presented by A.T. at the conference Interactions between Geometry and Physics — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).
Interactions between Geometry and Physics. A Special Issue in Honor of Ugo Bruzzo’s 60th Birthday
Keywords: AGT; Donaldson invariants; Equivariant localization; Exact partition function; Supersymmetry; Virasoro conformal blocks

On the generalized Cheeger problem and an application to 2d strips - Pratelli, A.; Saracco, G. (Rev. Mat. Iberoam., 2017)

On the genesis of directional friction through bristle-like mediating elements - Paolo Gidoni; Antonio DeSimone (ESAIM: COCV, 2017)
We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.

Globally stable quasistatic evolution for strain gradient plasticity coupled with damage - Vito Crismale (Annali di Matematica Pura ed Applicata (1923 -), 2017)
We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material's degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016).

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates - Alessandro Michelangeli; Alessandro Olgiati (Journal of Nonlinear Mathematical Physics, 2017)
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl’s projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

Homotopically invisible singular curves - Andrei A. Agrachev; Francesco Boarotto; Antonio Lerario (Calculus of Variations and Partial Differential Equations, 2017)

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry - Francesco Boarotto; Antonio Lerario (Communications in Analysis and Geometry, 2017)

$hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes - Andrea Cangiani; Dong, Zhaonan; E.H. Georgoulis; Houston, Paul (SpringerBriefs in Mathematics, 2017)

$hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes - Andrea Cangiani; Dong, Zhaonan; E.H. Georgoulis (SIAM J. Sci. Comput., 2017)

The injectivity radius of Lie manifolds - Paolo Antonini; Guido De Philippis; Nicola Gigli (ArXiv e-prints, 2017)
We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive
Keywords: (58J40); 53C21; Mathematics - Differential Geometry

Integrability of dominated decompositions on three-dimensional manifolds - Stefano Luzzatto; Sina Türeli; Khadim Mbacke War (Ergodic Theory and Dynamical Systems, 2017)
We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

Integrable lifts for transitive Lie algebroids - Androulidakis, I.; Paolo Antonini (ArXiv e-prints, 2017)
Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid.
Keywords: 14F40; 58H05; Mathematics - Differential Geometry

Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes - Massimiliano Rossi; Giancarlo Cicconofri; Alfred Beran; Giovanni Noselli; Antonio DeSimone (Proceedings of the National Academy of Sciences, 2017)
Active flagella provide the propulsion mechanism for a large variety of swimming eukaryotic microorganisms, from protists to sperm cells. Planar and helical beating patterns of these structures are recurrent and widely studied. The fast spinning motion of the locomotory flagellum of the alga Euglena gracilis constitutes a remarkable exception to these patterns. We report a quantitative description of the 3D flagellar beating in swimming E. gracilis. Given their complexity, these shapes cannot be directly imaged with current microscopy techniques. We show how to overcome these limitations by developing a method to reconstruct in full the 3D kinematics of the cell from conventional 2D microscopy images, based on the exact characterization of the helical motion of the cell body.The flagellar swimming of euglenids, which are propelled by a single anterior flagellum, is characterized by a generalized helical motion. The 3D nature of this swimming motion, which lacks some of the symmetries enjoyed by more common model systems, and the complex flagellar beating shapes that power it make its quantitative description challenging. In this work, we provide a quantitative, 3D, highly resolved reconstruction of the swimming trajectories and flagellar shapes of specimens of Euglena gracilis. We achieved this task by using high-speed 2D image recordings taken with a conventional inverted microscope combined with a precise characterization of the helical motion of the cell body to lift the 2D data to 3D trajectories. The propulsion mechanism is discussed. Our results constitute a basis for future biophysical research on a relatively unexplored type of eukaryotic flagellar movement.

The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes' phenomenon - Marco Bertola; Mattia Cafasso (Comm. Math. Phys, 2017)

Krein-Visik-Birman self-adjoint extension theory revisited - Matteo Gallone; Alessandro Michelangeli; Andrea Ottolini (2017)
The core results of the so-called KreIn-Visik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators are reproduced, both in their original and in a more modern formulation, within a comprehensive discussion that includes missing details, elucidative steps, and intermediate results of independent interest.

A Lagrangian approach for scalar multi-d conservation laws - Stefano Bianchini; Paolo Bonicatto; Elio Marconi (2017)

Lagrangian representations for linear and nonlinear transport - Stefano Bianchini; Paolo Bonicatto; Elio Marconi (Contemporary Mathematics. Fundamental Directions, 2017)
In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish - Riccardo Scala (ESAIM: COCV, 2017)
We introduce a model of dynamic evolution of a delaminated visco-elastic body with viscous adhesive. We prove the existence of solutions of the corresponding system of PDEs and then study the behavior of such solutions when the data of the problem vary slowly. We prove that a rescaled version of the dynamic evolutions converge to a “local” quasistatic evolution, which is an evolution satisfying an energy inequality and a momentum balance at all times. In the one-dimensional case we give a more detailed description of the limit evolution and we show that it behaves in a very similar way to the limit of the solutions of the dynamic model in [T. Roubicek, SIAM J. Math. Anal. 45 (2013) 101–126], where no viscosity in the adhesive is taken into account.

Linear Hyperbolic Systems in Domains with Growing Cracks - Maicol Caponi (Milan Journal of Mathematics, 2017)
We consider the hyperbolic system ü$${ - {\rm div} (\mathbb{A} \nabla u) = f}$$in the time varying cracked domain $${Ω \backslash Γ_t}$$, where the set $${Ω \subset \mathbb{R}^d}$$is open, bounded, and with Lipschitz boundary, the cracks $${Γ_t, t \in [0, T]}$$, are closed subsets of $${\bar{Ω}}$$, increasing with respect to inclusion, and $${u(t) : Ω \backslash Γ_t → \mathbb{R}^d}$$for every $${t \in [0, T]}$$. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈$${ - {\rm div} (\mathbb{B}\nabla v) + a\nabla v - 2 \nabla \dot{v}b = g}$$on the fixed domain $${Ω \backslash Γ_0}$$. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.

Linearisation of multiwell energies - Roberto Alicandro; Gianni Dal Maso; Giuliano Lazzaroni; Mariapia Palombaro (2017)
Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation - Gianni Dal Maso; Gianluca Orlando; Rodica Toader (Advances in Calculus of Variations, 2017)
We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to $L^1$ convergence.

A lower semicontinuity result for a free discontinuity functional with a boundary term - Stefano Almi; Gianni Dal Maso; Rodica Toader (Journal de Mathématiques Pures et Appliquées, 2017)
We study the lower semicontinuity in $GSBV^{p}(Ω;\mathbb{R}^{m})$ of a free discontinuity functional $\mathcal{F}(u)$ that can be written as the sum of a crack term, depending only on the jump set $S_{u}$, and of a boundary term, depending on the trace of $u$ on $\partialΩ$. We give sufficient conditions on the integrands for the lower semicontinuity of $\mathcal{F}$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of $\mathcal{F}$ can be represented by the sum of two integrals on $S_{u}$ and $\partialΩ$, respectively.

The Malgrange form and Fredholm determinants - Marco Bertola (SIGMA Symmetry Integrability Geom. Methods Appl., 2017)

Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation - Marco Bertola; Alexander Tovbis (Comm. Math. Phys., 2017)

Mean-field quantum dynamics for a mixture of Bose–Einstein condensates - Alessandro Michelangeli; Alessandro Olgiati (Analysis and Mathematical Physics, 2017)
We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schrödinger dynamics is effectively described by a system of coupled cubic non-linear Schrödinger equations, one for each component.

Minimizers of anisotropic perimeters with cylindrical norms - Giovanni Bellettini; Matteo Novaga; Shokhrukh Kholmatov (Communications on Pure & Applied Analysis, 2017)
We study various regularity properties of minimizers of the Φ–perimeter, where Φ is a norm. Under suitable assumptions on Φ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.
Keywords: anisotropic Bernstein problem; minimal cones; Non parametric minimal surfaces; Sets of finite perimeter

Model Reduction Methods - Francisco Chinesta; Antonio Huerta; Gianluigi Rozza; Karen Willcox (Encyclopedia of Computational Mechanics Second Edition, 2017)
This chapter presents an overview of model order reduction – a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control – critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1&isAllowed=y

Moduli of semistable sheaves as quiver moduli - Andrea Maiorana (2017)
In the 1980s Drézet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on P2 as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on P1×P1.

Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities - Guglielmo Feltrin (Communications on Pure & Applied Analysis, 2017)
We study the second order nonlinear differential equation \begindocument $ u'' + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocument where $\alpha_i, \beta_j>0$, $a_i(x), b_j(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen[0, L\mathclose]$, and the nonlinearities $g_i(s), k_j(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.p = 0$, with $p>1$.When the positive parameters $\beta_j$ are sufficiently large, we prove the existence of at least $2.m-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.
Keywords: Leray-Schauder topological degree; positive solutions; Sturm-Liouville boundary conditions; Superlinear indefinite problems

Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree - Guglielmo Feltrin; Fabio Zanolin (Journal of Differential Equations, 2017)
We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.
Keywords: Coincidence degree; Multiplicity results; Neumann boundary value problems; Positive periodic solutions; subharmonic solutions; Superlinear indefinite problems

A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling - Luca Heltai; Kiendl, J.; Antonio DeSimone; Alessandro Reali (COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2017)

A note on a fixed point theorem on topological cylinders - Guglielmo Feltrin (Ann. Mat. Pura Appl., 2017)
We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result, we discuss the existence of fixed points for maps defined on different types of domains and we propose alternative proofs for classical fixed point theorems, as Brouwer, Schauder and Krasnosel’skii ones.
AMS Subject Classification: 47H10, 37C25, 47H11, 54H25.

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations - Lorenzo Nardini (Journal of Dynamics and Differential Equations, 2017)

Numerical approximation of space-time fractional parabolic equations - Bonito, Andrea; Wenyu Lei; Joseph E Pasciak (Comput. Methods Appl. Math., 2017)

Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts - F. Ballarin; Elena Faggiano; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza; Sonia Ippolito; Roberto Scrofani (Biomechanics and Modeling in Mechanobiology, 2017)

POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder - Giovanni Stabile; Saddam Hijazi; Andrea Mola; Stefano Lorenzi; Gianluigi Rozza (Communications in Applied and Industrial Mathematics, 2017)
Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

A posteriori error estimates for the virtual element method - Andrea Cangiani; E.H. Georgoulis; Pryer, Tristan; Sutton, Oliver J. (Numer. Math., 2017)

Quasi-periodic solutions for quasi-linear generalized KdV equations - Filippo Giuliani (Journal of Differential Equations, 2017)
We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash–Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.
Keywords: KAM for PDE's; KdV; Nash–Moser theory; Quasi-linear PDE's; Quasi-periodic solutions

Quasi-periodic water waves - Massimiliano Berti; Montalto, Riccardo (J. Fixed Point Theory Appl., 2017)

Quasistatic crack growth based on viscous approximation: a model with branching and kinking - Vito Crismale; Giuliano Lazzaroni (Nonlinear Differential Equations and Applications NoDEA, 2017)
Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

Random spectrahedra - Paul Breiding; Khazhgali Kozhasov; Antonio Lerario (2017)

Rayleigh–Taylor instability in soft elastic layers - Davide Riccobelli; Pasquale Ciarletta (Phil. Trans. R. Soc. A, 2017)

Real topological string amplitudes - Narain, K. S.; Nicolò Piazzalunga; Alessandro Tanzini (Journal of High Energy Physics, 2017)
We discuss the physical superstring correlation functions in type I theory (or equivalently type II with orientifold) that compute real topological string amplitudes. We consider the correlator corresponding to holomorphic derivative of the real topological amplitude $\mathcal{G_χ}$, at fixed worldsheet Euler characteristic $χ$. This corresponds in the low-energy effective action to $\mathcal{N}=2$ Weyl multiplet, appropriately reduced to the orientifold invariant part, and raised to the power $g'= −χ+ 1$. We show that the physical string correlator gives precisely the holomorphic derivative of topological amplitude. Finally, we apply this method to the standard closed oriented case as well, and prove a similar statement for the topological amplitude $\mathcal{F}_g$.

Reduced Basis Methods for Uncertainty Quantification - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (SIAM/ASA Journal on Uncertainty Quantification, 2017)
In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined. Read More: http://epubs.siam.org/doi/abs/10.1137/151004550

A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool - Stefano Lorenzi; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (Applied Mathematical Modelling, 2017)

Reduced-order semi-implicit schemes for fluid-structure interaction problems - F. Ballarin; Gianluigi Rozza; Yvon Maday (Model Reduction of Parametrized Systems, 2017)
POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

Regularity estimates for scalar conservation laws in one space dimension - Elio Marconi (2017)
In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function ƒ has on the entropy solution. More precisely, if the set ⟨w : ƒ " (w) ≠ 0⟩ is dense, the regularity of the solution can be expressed in terms of BV Ф spaces, where Ф depends on the nonlinearity of ƒ. If moreover the set ⟨w : ƒ " (w) = 0⟩ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that ƒ' 0 u(t) ∈ BVloc (R) for every t > 0 and that this can be improved to SBVloc (R) regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian - Alessandro Olgiati (Advances in Quantum Mechanics: Contemporary Trends and Open Problems, 2017)
The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the ``projection counting'' scheme.

Second order differentiation formula on compact RCD*(K,N) spaces - Nicola Gigli; Luca Tamanini (2017)

Self-Adjoint Extensions of Dirac Operator with Coulomb Potential - Matteo Gallone (SISSA, 2017)
In this note we give a concise review of the present state-of-art for the problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular scalar potential V(x) = Ø(x)I4. We try to follow the historical and conceptual path that leads to the present understanding of the problem and to highlight the techniques employed and the main ideas. In the final part we outline a few major open questions that concern the topical problem of the multiplicity of self-adjoint realisations of the model, and which are worth addressing in the future.

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei - Matteo Gallone; Alessandro Michelangeli (2017)
We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the KreĬn-Višik- Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neu- mann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the exten- sion, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

Semistable Higgs Bundles on Calabi-Yau Manifolds - Ugo Bruzzo; Valeriano Lanza; Alessio Lo Giudice (2017)
We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifold. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.

Small Time Asymptotics on the Diagonal for Hörmander's Type Hypoelliptic Operators - Elisa Paoli (Journal of Dynamical and Control Systems, 2017)
We compute the small time asymptotics of the fundamental solution of Hörmander's type hypoelliptic operators with drift, on the diagonal at a point x0. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x0, then so is also the original control problem, and in this case we show that the fundamental solution blows up as t−N/2\$\backslashphantom {\backslashdot {i}\backslash!}t^{-\backslashmathcal {N}/2}\$, where N\$\backslashphantom {\backslashdot {i}\backslash!}\backslashmathcal {N}\$is a number determined by the Lie algebra at x0 of the fields, that define the hypoelliptic operator.

Solid tumors are poroelastic solids with a chemo-mechanical feedback on growth - D. Ambrosi; Pezzuto, S.; Davide Riccobelli; Stylianopoulos, T.; Pasquale Ciarletta (J. Elast., 2017)

Spectral Properties of the 2+1 Fermionic Trimer with Contact Interactions - Simon Becker; Alessandro Michelangeli; Andrea Ottolini (SISSA, 2017)
We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.
Partially supported by the 2014-2017 MIUR-FIR grant \Cond-Math: Condensed Matter and Mathematical Physics" code RBFR13WAET (S.B., A.M., A.O.), by the DAAD International Trainership Programme (S.B.), and by a 2017 visiting research fellowship at the International Center for Mathematical Research CIRM, Trento (A.M.).

Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler - Paolo Gidoni; Antonio DeSimone (Meccanica, 2017)
We formulate and solve the locomotion problem for a bio-inspired crawler consisting of two active elastic segments (i.e., capable of changing their rest lengths), resting on three supports providing directional frictional interactions. The problem consists in finding the motion produced by a given, slow actuation history. By focusing on the tensions in the elastic segments, we show that the evolution laws for the system are entirely analogous to the flow rules of elasto-plasticity. In particular, sliding of the supports and hence motion cannot occur when the tensions are in the interior of certain convex regions (stasis domains), while support sliding (and hence motion) can only take place when the tensions are on the boundary of such regions (slip surfaces). We solve the locomotion problem explicitly in a few interesting examples. In particular, we show that, for a suitable range of the friction parameters, specific choices of the actuation strategy can lead to net displacements also in the direction of higher friction.

Symplectic geometry of the moduli space of projective structures in homological coordinates - Marco Bertola; Dmitry Korotkin; Chaya Norton (Inventiones Mathematicae, 2017)

A uniqueness result for the decomposition of vector fields in Rd - Stefano Bianchini; Paolo Bonicatto (SISSA, 2017)
Given a vector field $\rho (1,\b) \in L^1_\loc(\R^+\times \R^{d},\R^{d+1})$ such that $\dive_{t,x} (\rho (1,\b))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,\b) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to \R^d$, $\dot \gamma (t)= \b(t,\gamma(t))$. We give conditions in terms of a local structure of the representation $\eta$ on suitable sets in order to prove that there is a partition of $\R^{d+1}$ into disjoint trajectories $\wp_\a$, $\a \in \A$, such that the PDE \begin{equation*} \dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u \in L^\infty(\R^+\times \R^{d}), \end{equation*} can be disintegrated into a family of ODEs along $\wp_\a$ with measure r.h.s.. The decomposition $\wp_\a$ is essentially unique. We finally show that $\b \in L^1_t(\BV_x)_\loc$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $\BV$ vector fields.

Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles - Marco Bertola; Mattia Cafasso (Random Matrices Theory Appl., 2017)

Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation - Tikan, Alexey; Billet, Cyril; Gennady El; Alexander Tovbis; Marco Bertola; Sylvestre, Thibaut; Gustave, Francois; Randoux, Stephane; Genty, Goëry; Suret, Pierre; Dudley, John M. (Phys. Rev. Lett., 2017)

Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations - Andrea Mola; Luca Heltai; Antonio DeSimone (Journal of Ship Research, 2017)
We present a model for the fast evaluation of the total drag of ship hulls operating in both wet and dry transom stern conditions, in calm or wavy water, based on the combination of an unsteady semi-Lagrangian potential flow formulation with fully nonlinear free-surface treatment, experimental correlations, and simplified viscous drag modeling. The implementation is entirely based on open source libraries. The spatial discretization is solved using a streamline upwind Petrov‐Galerkin stabilization of an iso-parametric, collocation based, boundary element method, implemented using the open source library deal.II. The resulting nonlinear differential-algebraic system is integrated in time using implicit backward differentiation formulas, implemented in the open source library SUNDIALS. The Open CASCADE library is used to interface the model directly with computer-aided design data structures. The model accounts automatically for hulls with a transom stern, both in wet and dry regimes, by using a specific treatment of the free-surface nodes on the stern edge that automatically detects when the hull advances at low speeds. In this case, the transom stern is partially immersed, and a pressure patch is applied on the water surface detaching from the transom stern, to recover the gravity effect of the recirculating water on the underlying irrotational flow domain. The parameters of the model used to impose the pressure patch are approximated from experimental relations found in the literature. The test cases considered are those of the U.S. Navy Combatant DTMB-5415 and the National Physical Laboratory hull. Comparisons with experimental data on quasi-steady test cases for both water elevation and total hull drag are presented and discussed. The quality of the results obtained on quasi-steady simulations suggests that this model can represent a promising alternative to current unsteady solvers for simulations with Froude numbers below 0.35.

Adaptivity and blow-up detection for nonlinear evolution problems - Andrea Cangiani; E.H. Georgoulis; Kyza, Irene; Metcalfe, Stephen (SIAM J. Sci. Comput., 2016)

Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives - Filippo Salmoiraghi; F. Ballarin; Giovanni Corsi; Andrea Mola; Marco Tezzele; Gianluigi Rozza (Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering,, 2016)
Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.

On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity - Giovanni Bellettini; Lucia Tealdi; Maurizio Paolini (ESAIM: COCV, 2016)
In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau's problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of this surface, namely a conformal parametrization defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some result from Morse theory.
Keywords: Area functional

On asymptotic regimes of orthogonal polynomials with complex varying quartic exponential weight - Marco Bertola; Alexander Tovbis (SIGMA Symmetry Integrability Geom. Methods Appl., 2016)

Behaviour of the reference measure on RCD spaces under charts - Nicola Gigli; Enrico Pasqualetto (2016)

Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers - Giordano Cotti (2016)

Comparing Poisson Sigma Model with A-model - Bonechi, F.; Cattaneo, A.S.; Riccardo Iraso (Journal of High Energy Physics, 2016)
We discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the discussion in [4], where a gauge fixing defined by a compatible complex structure was introduced, by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss the off-shell supersymmetry of the A-model as a residual BV symmetry of the gauge fixed PSM action.

On the concentration of entropy for scalar conservation laws - Stefano Bianchini; Elio Marconi (Discrete & Continuous Dynamical Systems - S, 2016)
We prove that the entropy for an $L^∞$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution.
Keywords: concentration; Conservation laws; entropy solutions; Lagrangian representation; shocks

Confinement of dislocations inside a crystal with a prescribed external strain - Ilaria Lucardesi; Marco Morandotti; Riccardo Scala; Davide Zucco (2016)
We study screw dislocations in an isotropic crystal undergoing antiplane shear. In the framework of linear elasticity, by fixing a suitable boundary condition for the strain (prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Ƭ-convergence theory, in the framework of the so-called core radius approach.
Preprint SISSA 20/2016/MATE

Conformal Equivalence of 3D Contact Structures on Lie Groups - Francesco Boarotto (Journal of Dynamical and Control Systems, 2016)
In this paper, a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular, we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $mathbbH^3$ or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to $SU(2,1)$.

Construction of Real-Valued Localized Composite Wannier Functions for Insulators - Domenico Fiorenza; Domenico Monaco; Gianluca Panati (Annales Henri Poincaré, 2016)
We consider a real periodic Schrödinger operator and a physically relevant family of $m ≥ 1$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $d≤ 3$, there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.

Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$ - Marco Bertola; Boris Dubrovin; Di Yang (Phys. D, 2016)

CORRIGENDUM: The dependence on the monodromy data of the isomonodromic tau function - Marco Bertola (2016)

Critical points of a perturbed Otha-Kawasaki functional - Matteo Rizzi (arXiv preprint arXiv:1601.07093, 2016)

Currents and dislocations at the continuum scale - Riccardo Scala; Nicolas Van Goethem (Methods and Applications of Analysis, 2016)
A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in $L^p$ with $1≤ p<2$, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

The deal.II Library, Version 8.3 - Wolfgang Bangerth; Timo Heister; Luca Heltai; G. Kanschat; Martin Kronbichler; Matthias Maier; Bruno Turcksin (ARCHIVE OF NUMERICAL SOFTWARE, 2016)

The deal.II library, Version 8.4 - Wolfgang Bangerth; Denis Davydov; Timo Heister; Luca Heltai; G. Kanschat; Martin Kronbichler; Matthias Maier; Bruno Turcksin; David Wells (JOURNAL OF NUMERICAL MATHEMATICS, 2016)

Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes - Andrea Cangiani; E.H. Georgoulis; Jensen, Max (Appl. Numer. Math., 2016)

Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces - Nicola Gigli; Enrico Pasqualetto (2016)

Error Estimates of B-spline based finite-element method for the wind-driven ocean circulation - Rotundo, N.; Kim, T. -Y.; Jiang, W.; Luca Heltai; Fried, E. (JOURNAL OF SCIENTIFIC COMPUTING, 2016)

Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I - Giovanni Alberti; Stefano Bianchini; Laura Caravenna (Journal of Differential Equations, vol. 261, issue 8 (2016): 4298-4337, 2016)

Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux II - Giovanni Alberti; Stefano Bianchini; Laura Caravenna (2016)

Exact results for N=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants - Mikhail Bershtein; Giulio Bonelli; Massimiliano Ronzani; Alessandro Tanzini (Journal of High Energy Physics, 2016)
We provide a contour integral formula for the exact partition function of $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)\; \mathcal{N}=2^\star$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the $\mathcal{N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces - Luca Battaglia; Andrea Malchiodi (Journal of Functional Analysis, 2016)
We consider the SU(3) singular Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑m=1Mα1m(δpm−1)−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑m=1Mα2m(δpm−1), where hi are smooth positive functions on Σ, ρi∈R+, pm∈Σ and αim>−1. We give both existence and non-existence results under some conditions on the parameters ρi and αim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities."
Keywords: Liouville-type equations; Min–max solutions; Non-existence results; Toda system

Existence and uniqueness of dynamic evolutions for a peeling test in dimension one - Gianni Dal Maso; Giuliano Lazzaroni; Lorenzo Nardini (Journal of Differential Equations, 2016)
In this paper we present a one-dimensional model of a dynamic peeling test for a thin film, where the wave equation is coupled with a Griffith criterion for the propagation of the debonding front. Our main results provide existence and uniqueness for the solution to this coupled problem under different assumptions on the data.
Keywords: Dynamic debonding; Dynamic energy release rate; Dynamic fracture; Griffith's criterion; Maximum dissipation principle; Wave equation in time-dependent domains

A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts - F. Ballarin; Elena Faggiano; Andrea Manzoni; Gianluigi Rozza; Alfio Quarteroni; Sonia Ippolito; Roberto Scrofani; Carlo Antona (Submitted, 2016)
A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.

Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case - Gianni Dal Maso; Gianluca Orlando; Rodica Toader (Calculus of Variations and Partial Differential Equations, 2016)
We study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $ε$, which forces damage concentration on regions of codimension one. We determine the $Γ$-limit as $ε$ tends to zero and show that it contains an energy term involving the crack opening.

Frames symplectic sheaves on surfaces and their ADHM data - Jacopo Vittorio Scalise (SISSA, 2016)
This dissertation is centered on the moduli space of what we call framed symplectic sheaves on a surface, compactifying the corresponding moduli space of framed principal SP−bundles. It contains the construction of the moduli space, which is carried out for every smooth projective surface X with a big and nef framing divisor, and a study of its deformation theory. We also develop an in-depth analysis of the examples X = P2 and X = Blp (P2 ), showing that the corresponding moduli spaces enjoy an ADHM-type description. In the former case, we prove irreducibility of the space and exhibit a relation with the space of framed ideal instantons on S4 in type C.
Keywords: moduli spaces

A Frobenius theorem for corank-1 continuous distributions in dimensions two and three - Stefano Luzzatto; Sina Türeli; Khadim Mbacke War (International Journal of Mathematics, 2016)
We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive C1 distribution is uniquely integrable.

Generalizing the Poincaré–Miranda theorem: the avoiding cones condition - Alessandro Fonda; Paolo Gidoni (Annali di Matematica Pura ed Applicata (1923 -), 2016)
After proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \$\$\backslashpm \$\$±1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.

Globally stable quasistatic evolution for a coupled elastoplastic–damage model - Vito Crismale (ESAIM: Control, Optimisation and Calculus of Variations, 2016)
We show the existence of globally stable quasistatic evolutions for a rate-independent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage.

The Gysin sequence for quantum lens spaces - Francesca Arici; Simon Brain; Giovanni Landi (Journal of Noncommutative Geometry, 2016)
We define quantum lens spaces as ‘direct sums of line bundles’ and exhibit them as ‘total spaces’ of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as ‘line bundles’ over quantum lens spaces and generically define ‘torsion classes’. We work out explicit examples of these classes.

Hankel determinant approach to generalized Vorob'ev-Yablonski polynomials and their roots - Ferenc Balogh; Marco Bertola; Thomas Bothner (Constr. Approx., 2016)

$hp$-version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes - Andrea Cangiani; Dong, Zhaonan; E.H. Georgoulis; Houston, Paul (ESAIM Math. Model. Numer. Anal., 2016)

Instanton counting on compact manifolds - Massimiliano Ronzani (SISSA, 2016)
In this thesis we analyze supersymmetric gauge theories on compact manifolds and their relation with representation theory of infinite Lie algebras associated to conformal field theories, and with the computation of geometric invariants and superconformal indices. The thesis contains the work done by the candidate during the doctorate programme at SISSA under the supervision of A. Tanzini and G. Bonelli. • in Chapter 2, we consider N = 2 supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity. • in Chapter 3, we provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2∗ theory on P2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. • in Chapter 4, we explore N = (1, 0) superconformal six-dimensional theories arising from M5 branes probing a transverse Ak singularity. Upon circle compactification to five dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional in- stanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show which relations among vertex correlators of qW algebrae are implied by the S-duality of the pq-web.
Keywords: Supersymmetry

Integrability of C1 invariant splittings - Stefano Luzzatto; Sina Türeli; Khadim Mbacke War (Dynamical Systems, 2016)
We derive some new conditions for integrability of dynamically defined C1 invariant splittings, formulated in terms of the singular values of the iterates of the derivative of the diffeomorphism which defines the splitting.

Integrability of continuous bundles and applications to dynamical systems - Khadim Mbacke War (SISSA, 2016)
In this dissertation we study the problem of integrability of bundles with low regularities.

Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes - Filippo Salmoiraghi; F. Ballarin; Luca Heltai; Gianluigi Rozza (Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences, 2016)
In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model.

KAM for autonomous quasi-linear perturbations of KdV - Baldi, Pietro; Massimiliano Berti; Montalto, Riccardo (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2016)

KAM for autonomous quasi-linear perturbations of mKdV - Baldi, Pietro; Massimiliano Berti; Montalto, Riccardo (Boll. Unione Mat. Ital., 2016)

KAM for PDEs - Massimiliano Berti (Boll. Unione Mat. Ital., 2016)

LinearOperator – a generic, high-level expression syntax for linear algebra - Matthias Maier; Mauro Bardelloni; Luca Heltai (COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2016)

A model for the quasistatic growth of cracks with fractional dimension - Gianni Dal Maso; Marco Morandotti (2016)
We study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated.

Model Order Reduction: a survey - Francisco Chinesta; Antonio Huerta; Gianluigi Rozza; Karen Willcox (Wiley Encyclopedia of Computational Mechanics, 2016, 2016)

Moser–Trudinger inequalities for singular Liouville systems - Luca Battaglia (Mathematische Zeitschrift, 2016)
In this paper we prove a Moser–Trudinger inequality for the Euler–Lagrange functional of general singular Liouville systems on a compact surface. We characterize the values of the parameters which yield coercivity for the functional, hence the existence of energy-minimizing solutions for the system, and we give necessary conditions for boundedness from below. We also provide a sharp inequality under assuming the coefficients of the system to be non-positive outside the diagonal. The proofs use a concentration-compactness alternative, Pohožaev-type identities and blow-up analysis.

Motion planning and motility maps for flagellar microswimmers - Giancarlo Cicconofri; Antonio DeSimone (The European Physical Journal E, 2016)
We study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity-modulated inputs, the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas, whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations.

A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel - Alberto Sartori; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208, 2016)
In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well.

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type - Alessandro Michelangeli; Andrea Ottolini (2016)
We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan- Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Kreĭn, Višiik, and Birman. We identify the explicit `Kreĭn-Višik-Birman extension param- eter' as an operator on the `space of charges' for this model (the `Kreĭn space') and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we re- produce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.

New existence results for the mean field equation on compact surfaces via degree theory - Aleks Jevnikar (Rend. Sem. Mat. Univ. Padova, 2016)

The nonconforming virtual element method for the Stokes equations - Andrea Cangiani; Gyrya, Vitaliy; Manzini, Gianmarco (SIAM J. Numer. Anal., 2016)

Non-linear Schrödinger system for the dynamics of a binary condensate: theory and 2D numerics - Alessandro Michelangeli; Giuseppe Pitton (2016)
We present a comprehensive discussion of the mathematical framework for binary Bose-Einstein condensates and for the rigorous derivation of their effective dynamics, governed by a system of coupled non-linear Gross-Pitaevskii equations. We also develop in the 2D case a systematic numerical study of the Gross-Pitaevskii systems in a wide range of relevant regimes of population ratios and intra-species and inter-species interactions. Our numerical method is based on a Fourier collocation scheme in space combined with a fourth order integrating factor scheme in time.

A note on a multiplicity result for the mean field equation on compact surfaces - Aleks Jevnikar (Advanced Nonlinear Studies, 2016)

Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case - Alberto Boscaggin; Guglielmo Feltrin; Fabio Zanolin (Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474., 2016)
We study the periodic and Neumann boundary value problems associated with the second order nonlinear differential equation u''+cu'+λa(t)g(u)=0, where g:[0,+∞[→[0,+∞[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when ∫a(t)dt 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.
AMS Subject Classification: Primary 34B18; 34C25; Secondary 34B15; 47H11;

Periodic perturbations of Hamiltonian systems - Alessandro Fonda; Maurizio Garrione; Paolo Gidoni (Advances in Nonlinear Analysis, 2016)
We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

Pimsner Algebras and Circle Bundles - Francesca Arici; Francesco D'Andrea; Giovanni Landi (Noncommutative Analysis, Operator Theory and Applications, 2016)
We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with examples of quantum weighted projective and lens spaces and θ-deformations.

Pimsner algebras and Gysin sequences from principal circle actions - Francesca Arici; Jens Kaad; Giovanni Landi (Journal of Noncommutative Geometry, 2016)

POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations - Stefano Lorenzi; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, Elsevier, 2016)
Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition – Reduced Order Model (POD – ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re = 103 and Re = 105 have been considered in order to assess both a laminar and turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.

POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems - F. Ballarin; Gianluigi Rozza (International Journal Numerical Methods for Fluids, 2016)
In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline-online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic POD–Galerkin method for the online computation of the global structural displacement, fluid velocity and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced order method and its computational performances

On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians - Alessandro Michelangeli; Andrea Ottolini (2016)
For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the ``Ter-Martirosyan--Skornyakov condition'' gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Krein, Visik, and Birman. We show that the Ter-Martirosyan--Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a point-wise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature.

Positive solutions to indefinite problems: a topological approach - Guglielmo Feltrin (SISSA, 2016)
The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by Gómez‐Reñasco and López‐Gómez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + λ a(t) g(u) = 0, when both λ>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions. As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations.
The research work described in this Ph.D. thesis has produced 10 papers.
Keywords: positive solutions

A quadratic interaction estimate for conservation laws: motivations, techniques and open problems - Stefano Modena (Bulletin of the Brazilian Mathematical Society, New Series, 2016)
In a series of joint works with S. Bianchini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of this paper is to present the results obtained in the three cited articles [3, 4, 5], discussing how they are related with the general theory of hyperbolic conservation laws. To this purpose, first we explain why this quadratic estimate is interesting, then we give a brief overview of the techniques we used to prove it and finally we present some related open problems.

Quadratic interaction estimate for hyperbolic conservation laws, an overview - Stefano Modena (Contemporary Mathematics. Fundamental Directions, 2016)

Qualitative properties and construction of solutions to some semilinear elliptic PDEs - Matteo Rizzi (SISSA, 2016)
This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction.
Keywords: moving planes method, maximum principle, Lyapunov-Schmidt reduction, Willmore surfaces, Otha-Kawasaki functional

Quasi-static hydraulic crack growth driven by Darcy's law - Stefano Almi (2016)
In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain ε ∈ R and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V , and ε. Then, we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods - Alberto Sartori; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (NERS-14-1062; ASME J of Nuclear Rad Sci, 2, 2 (2016) 021019, 2016)
This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computational time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased.
8 pages

Reduced basis approaches in time-dependent noncoercive settings for modelling the movement of nuclear reactor control rods - Alberto Sartori; Antonio Cammi; Lelio Luzzi; Gianluigi Rozza (Communications in Computational Physics, 2016)
In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries - Laura Iapichino; Alfio Quarteroni; Gianluigi Rozza (Computers and Mathematics with Applications, 2016)
The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed.

Refined node polynomials via long edge graphs - Lothar Göttsche; Benjamin Kipkirui Kikwai (Communications in Number Theory and Physics, 2016)
The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for toric surfaces by Block, Colley, Kennedy and Liu, Osserman, using tropical geometry and in particular the combinatorial tool of long-edged graphs. In the first part of this paper these results are for $\mathbb{P}^2$ and rational ruled surfaces generalised to refined Severi degrees. In the second part of the paper we give a number of mostly conjectural generalisations of this result to singular surfaces, and curves with prescribed multiple points. The formulas involve modular forms and theta functions.

Renormalization for Autonomous Nearly Incompressible BV Vector Fields in Two Dimensions - Stefano Bianchini; Paolo Bonicatto; N.A. Gusev (SIAM Journal on Mathematical Analysis, 2016)
Given a bounded autonomous vector field $b \colon \mathbb{R}^d → \mathbb{R}^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b · \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].

Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains - Antonietti, Paola F.; Andrea Cangiani; Collis, Joe; Dong, Zhaonan; E.H. Georgoulis; Stefano Giani; Houston, Paul (Building bridges: connections and challenges in modern approaches to numerical partial differential equations, 2016)

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation - Marco Bertola; Gennady El; Alexander Tovbis (Proc. A., 2016)

Second-order structured deformations - Ana Cristina Barroso; Jose Matias; Marco Morandotti; David R. Owen (SISSA, 2016)

Ship Sinkage and Trim Predictions Based on a CAD Interfaced Fully Nonlinear Potential Model - Andrea Mola; Luca Heltai; Antonio DeSimone (The 26th International Ocean and Polar Engineering Conference, 2016)

Simple Lie Algebras and Topological ODEs - Marco Bertola; Boris Dubrovin; Di Yang (Int. Math. Res. Not., 2016)

On Sobolev instability of the interior problem of tomography - Marco Bertola; Alexander Katsevich; Alexander Tovbis (Journal of Mathematical Analysis and Applications, 2016)

Some results on quasistatic evolution problems for unidirectional processes - Vito Crismale (SISSA, 2016)

Some results on the mathematical analysis of crack problems with forces applied on the fracture lips - Stefano Almi (SISSA, 2016)
This thesis is devoted to the study of some models of fracture growth in elastic materials, characterized by the presence of forces acting on the crack lips. Working in the general framework of rate-independent processes, we first discuss a variational formulation of the problem of quasi-static crack evolution in hydraulic fracture. Then, we investigate the crack growth process in a cohesive fracture model, showing the existence of an evolution satisfying a weak Griffith's criterion. Finally, in the last chapter of this work we investigate, in the static case, the interaction between the energy spent in order to create a new fracture and the energy spent by the applied surface forces. This leads us to study the lower semicontinuity properties of a free discontinuity functional F(u) that can be written as the sum of a crack term, depending on the jump set of u, and of a boundary term, depending on the trace of u.
Keywords: Fracture mechanics

Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds - Ugo Boscain; Dario Prandi; M. Seri (Communications in Partial Differential Equations, 2016)
We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

On the structure of $L^∞$-entropy solutions to scalar conservation laws in one-space dimension - Stefano Bianchini; Elio Marconi (SISSA, 2016)
We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f''=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index - Benvenuti, Sergio; Giulio Bonelli; Massimiliano Ronzani; Alessandro Tanzini (Journal of High Energy Physics, 2016)
We explore $\mathcal{N}=(1,0)$ superconformal six-dimensional theories arising from M5 branes probing a transverse $A_k$ singularity. Upon circle compactification to 5 dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional instanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show that S-duality of the pq-web implies new relations among vertex correlators of $q\mathcal{W}$ algebrae.

Symmetry properties of some solutions to some semilinear elliptic equations - Farina, Alberto; Andrea Malchiodi; Matteo Rizzi (Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 2016)