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} }