@article {2014, title = {Maximal generalized solution of eikonal equation}, number = {Journal of differential equations;volume 257; issue 1; pages 231-263;}, year = {2014}, publisher = {Elsevier}, abstract = {We study the Dirichlet problem for the eikonal equation: 1/2 |∇u(x)|^2-a(x)=0 in Ω u(x)=(x) on Ω, without continuity assumptions on the map a(.). We find a class of maps a(.) contained in the space L$\infty$(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.}, doi = {10.1016/j.jde.2014.04.001}, url = {http://urania.sissa.it/xmlui/handle/1963/34642}, author = {Sandro Zagatti} } @article {2011, title = {Compactness by maximality}, year = {2011}, abstract = {We derive a compactness property in the Sobolev space $W^{1,1}(\O)$ in order to study the Dirichlet problem for the eikonal equation \begin{displaymath} \begin{cases} \ha |\n u(x)|^2 - a(x) = 0 \& \ \textrm{in} \ \O\cr u(x)=\varphi(x) \& \ \textrm{on} \ \partial \O, \end{cases} \end{displaymath} without continuity assumptions on the map $a$.}, url = {http://preprints.sissa.it/handle/1963/35317}, author = {Sandro Zagatti} } @article {2011, title = {An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations}, journal = {Journal of Convex Analysis 18 (2011) 1141-1170}, year = {2011}, publisher = {Heldermann Verlag}, abstract = {We devote the \\\\textit{integro-extremization} method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations \\\\begin{displaymath} \\\\begin{cases} F(Du)=0 \& \\\\quad \\\\textrm{in} \\\\quad\\\\O\\\\cr u(x)=\\\\varphi(x) \& \\\\quad \\\\textrm{for} \\\\quad x\\\\in \\\\partial \\\\O, \\\\end{cases} \\\\end{displaymath} with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations \\\\begin{displaymath} \\\\begin{cases} \\\\frac{\\\\partial u}{\\\\partial t}+ F(\\\\nabla u)=0 \& \\\\quad \\\\textrm{in} \\\\quad ]0,T[\\\\times \\\\O\\\\cr u(0,x)=\\\\eta(x) \& \\\\quad \\\\textrm{for} \\\\quad x\\\\in\\\\O \\\\cr u(t,x)=\\\\psi(x) \& \\\\quad \\\\textrm{for} \\\\quad (t,x)\\\\in[0,T]\\\\times \\\\partial \\\\O. \\\\end{cases} \\\\end{displaymath} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\\\\O$ and on the boundary datum, without any growth assumptions on $F$.}, url = {http://hdl.handle.net/1963/5538}, author = {Sandro Zagatti} } @article {2009, title = {On viscosity solutions of Hamilton-Jacobi equations}, journal = {Trans. Amer. Math. Soc. 361 (2009) 41-59}, year = {2009}, publisher = {American Mathematical Society}, abstract = {We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.}, doi = {10.1090/S0002-9947-08-04557-1}, url = {http://hdl.handle.net/1963/3420}, author = {Sandro Zagatti} } @article {2008, title = {Minimization of non quasiconvex functionals by integro-extremization method}, journal = {Discrete Contin. Dyn. Syst. 21 (2008) 625-641}, year = {2008}, doi = {10.3934/dcds.2008.21.625}, url = {http://hdl.handle.net/1963/2761}, author = {Sandro Zagatti} } @article {2008, title = {Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations}, journal = {Calc. Var. Partial Differential Equations 31 (2008) 511-519}, year = {2008}, doi = {10.1007/s00526-007-0124-7}, url = {http://hdl.handle.net/1963/2760}, author = {Sandro Zagatti} } @article {2007, title = {Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals}, journal = {J. Math. Anal. Appl. 335 (2007) 1143-1160}, year = {2007}, abstract = {We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton-Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.}, doi = {10.1016/j.jmaa.2007.02.034}, url = {http://hdl.handle.net/1963/2763}, author = {Sandro Zagatti} } @article {2007, title = {Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient}, journal = {J. Convex Anal. 14 (2007) 705-727}, year = {2007}, abstract = {We study some qualitative properties of the integro-extremal minimizers of the functional of the gradient defined on Sobolev spaces with Dirichlet boundary conditions. We discuss their use in the non-convex case via viscosity methods and give conditions under which they are unique and depend continuously on boundary data.}, url = {http://hdl.handle.net/1963/2762}, author = {Sandro Zagatti} } @article {2005, title = {On the Minimum Problem for Nonconvex Scalar Functionals}, journal = {SIAM J. Math. Anal. 37 (2005) 982-995}, year = {2005}, abstract = {We study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.}, doi = {10.1137/040612506}, url = {http://hdl.handle.net/1963/2764}, author = {Sandro Zagatti} } @article {2000, title = {Minimization of functionals of the gradient by Baire{\textquoteright}s theorem}, journal = {SIAM J. Control Optim. 38 (2000) 384-399}, number = {SISSA;71/97/M}, year = {2000}, publisher = {SIAM}, abstract = {
We give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_\Omega g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(\Omega,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.
}, doi = {10.1137/S0363012998335206}, url = {http://hdl.handle.net/1963/3511}, author = {Sandro Zagatti} } @article {1998, title = {On the Dirichlet problem for vectorial Hamilton-Jacobi equations}, journal = {SIAM J. Math. Anal. 29 (1998) 1481-1491}, number = {SISSA;182/96/M}, year = {1998}, publisher = {SIAM}, abstract = {We give sufficient conditions for the existence of solutions to the Hamilton--Jacobi equations with Dirichlet boundary condition: $$ \\\\cases{ g(x,{\\\\hbox{\\\\rm det}}Du(x))=0, \\\\ \& for a.e. $x\\\\in\\\\Omega,$\\\\cr u(x)=\\\\varphi(x), \& for $x\\\\in\\\\partial\\\\Omega,$} $$ obtaining, in addition, an application to the theory of existence of minimizers for a class of nonconvex variational problems.}, doi = {10.1137/S0036141097321279}, url = {http://hdl.handle.net/1963/3512}, author = {Sandro Zagatti} } @article {1995, title = {An existence result in a problem of the vectorial case of the calculus of variations}, year = {1995}, publisher = {SIAM}, abstract = {SIAM J. Control Optim. 33 (1995) 960-970}, url = {http://hdl.handle.net/1963/3513}, author = {Arrigo Cellina and Sandro Zagatti} } @article {1994, title = {A version of Olech\\\'s lemma in a problem of the calculus of variations}, journal = {SIAM J. Control Optim. 32 (1994) 1114-1127}, year = {1994}, publisher = {SIAM}, abstract = {This paper studies the solutions of the minimum problem for a functional of the gradient under linear boundary conditions. A necessary and sufficient condition, based on the facial structure of the epigraph of the integrand, is provided for the continuous dependence of the solutions on boundary data.}, doi = {10.1137/S0363012992234669}, url = {http://hdl.handle.net/1963/3514}, author = {Arrigo Cellina and Sandro Zagatti} } @mastersthesis {1992, title = {Some Problems in the Calculus of the Variations}, year = {1992}, school = {SISSA}, keywords = {Calculus of variations}, url = {http://hdl.handle.net/1963/5428}, author = {Sandro Zagatti} }