01580nas a2200157   4500008004100000245009300041210006900134260002800203520103900231100001901270700001901289700002101308700002001329700002201349856005101371       2014                        en  d00aConformal invariants from nodal sets. I. negative eigenvalues and curvature prescription0 aConformal invariants from nodal sets I negative eigenvalues and   bOxford University Press3 aIn this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n≥3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n≥3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the Q-curvature prescription problems for noncritical Q-curvatures.1 aGover, Rod, R.1 aCanzani, Yaiza1 aJakobson, Dmitry1 aPonge, Raphaël1 aMalchiodi, Andrea  uhttp://urania.sissa.it/xmlui/handle/1963/3512801111nas a2200121   4500008004100000245006400041210006300105260003400168520070800202100002200910700002100932856003600953       2014                        en  d00aCritical points of the Moser-Trudinger functional on a disk0 aCritical points of the MoserTrudinger functional on a disk  bEuropean Mathematical Society3 aOn the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional
  $$E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$$ and its restrictions to
$M_\Lambda:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=\Lambda\}$ for $\Lambda>0$. We
prove that if a sequence $u_k$ of positive critical points of
$E|_{M_{\Lambda_k}}$ (for some $\Lambda_k>0$) blows up as $k\to\infty$, then
$\Lambda_k\to 4\pi$, and $u_k\to 0$ weakly in $H^1_0(B_1)$ and strongly in
$C^1_{\loc}(\bar B_1\setminus\{0\})$.
  Using this we also prove that when $\Lambda$ is large enough, then
$E|_{M_\Lambda}$ has no positive critical point, complementing previous
existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.1 aMalchiodi, Andrea1 aMartinazzi, Luca  uhttp://hdl.handle.net/1963/656001152nas a2200145   4500008004100000245009700041210006900138260001000207520067600217100002200893700001500915700002000930700002000950856003600970       2012                        en  d00aA Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.0 aCodazzilike equation and the singular set for C1 smooth surfaces  bSISSA3 aIn this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 11 aMalchiodi, Andrea1 aYang, Paul1 aCheng, Jih-Hsin1 aHwang, JennFang  uhttp://hdl.handle.net/1963/655600981nas a2200121   4500008004100000245006900041210006700110260001300177520058600190100002500776700002200801856003600823       2011                        en  d00aA class of existence results for the singular Liouville equation0 aclass of existence results for the singular Liouville equation  bElsevier3 aWe consider a class of elliptic PDEs on closed surfaces with exponential nonlinearities and Dirac deltas on the right-hand side. The study arises from abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities. A general existence result is proved using global variational methods: the analytic problem is reduced to a topological problem concerning the contractibility of a model space, the so-called space of formal barycenters, characterizing the very low sublevels of a suitable functional.1 aCarlotto, Alessandro1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/579300428nas a2200133   4500008004100000245005400041210005300095260001000148653003100158100002600189700002200215700002100237856003600258       2011                        en  d00aCritical points of the Moser-Trudinger functional0 aCritical points of the MoserTrudinger functional  bSISSA10aMoser-Trudinger inequality1 aDe Marchis, Francesca1 aMalchiodi, Andrea1 aMartinazzi, Luca  uhttp://hdl.handle.net/1963/459200912nas a2200133   4500008004300000245011700043210006900160520042800229100002600657700002200683700001900705700001800724856003600742       2010                        en_Ud  00aConcentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions0 aConcentration of solutions for some singularly perturbed mixed p3 aIn this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in.1 aGarcia Azorero, Jesus1 aMalchiodi, Andrea1 aMontoro, Luigi1 aPeral, Ireneo  uhttp://hdl.handle.net/1963/340900856nas a2200133   4500008004300000245010300043210006900146520038600215100002600601700002200627700001900649700001800668856003600686       2010                        en_Ud  00aConcentration of solutions for some singularly perturbed mixed problems. Part I: existence results0 aConcentration of solutions for some singularly perturbed mixed p3 aIn this paper we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.1 aGarcia Azorero, Jesus1 aMalchiodi, Andrea1 aMontoro, Luigi1 aPeral, Ireneo  uhttp://hdl.handle.net/1963/340600588nas a2200097   4500008004300000245007600043210006900119520024400188100002200432856003600454       2008                        en_Ud  00aConcentrating solutions of some singularly perturbed elliptic equations0 aConcentrating solutions of some singularly perturbed elliptic eq3 aWe study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/265700919nas a2200109   4500008004300000245008500043210006900128520053400197100002000731700002200751856003600773       2007                        en_Ud  00aConcentration on minimal submanifolds for a singularly perturbed Neumann problem0 aConcentration on minimal submanifolds for a singularly perturbed3 aWe consider the equation $- \\\\e^2 \\\\D u + u= u^p$ in $\\\\Omega \\\\subseteq \\\\R^N$, where $\\\\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\partial \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, ..., N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<p <\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/201300549nas a2200121   4500008004300000245010200043210007000145260003400215520009600249100002400345700002200369856003600391       2007                        en_Ud  00aConcentration phenomena for nonlinear Schrödinger equations: Recent results and new perspectives0 aConcentration phenomena for nonlinear Schrödinger equations Rece  bAmerican Mathematical Society3 aWe survey some results on (NLSepsilon), discussing also new perspectives and open problems.1 aAmbrosetti, Antonio1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/351600627nas a2200097   4500008004300000245004800043210004700091520033300138100002200471856003600493       2007                        en_Ud  00aConformal Metrics with Constant Q-Curvature0 aConformal Metrics with Constant QCurvature3 aWe consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/260500823nas a2200097   4500008004300000245007000043210006900113520048500182100002200667856003600689       2006                        en_Ud  00aCompactness of solutions to some geometric fourth-order equations0 aCompactness of solutions to some geometric fourthorder equations3 aWe prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/212600921nas a2200109   4500008004300000245009700043210006900140520052400209100002000733700002200753856003600775       2006                        en_Ud  00aConcentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem0 aConcentration at manifolds of arbitrary dimension for a singular3 aWe consider the equation $- \\\\e^2 \\\\D u + u = u^p$ in $\\\\O \\\\subseteq \\\\R^N$, where $\\\\O$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\pa \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, \\\\dots, N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/217000658nas a2200109   4500008004100000245010000041210006900141260001300210520026700223100002200490856003600512       2005                        en  d00aConcentration at curves for a singularly perturbed Neumann problem in three-dimensional domains0 aConcentration at curves for a singularly perturbed Neumann probl  bSpringer3 aWe prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain R3 and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/486600727nas a2200121   4500008004300000245006600043210006400109260003400173520032000207100002000527700002200547856003600569       2002                        en_Ud  00aCurvature theory of boundary phases: the two-dimensional case0 aCurvature theory of boundary phases the twodimensional case  bEuropean Mathematical Society3 aWe describe the behaviour of minimum problems involving non-convex surface integrals in 2D, singularly perturbed by a curvature term. We show that their limit is described by functionals which take into account energies concentrated on vertices of polygons. Non-locality and non-compactness effects are highlighted.1 aBraides, Andrea1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/3537