@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 = {<p>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 $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.</p>},
	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}
}
