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 $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.

PB - SIAM UR - http://hdl.handle.net/1963/3511 U1 - 753 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On the Dirichlet problem for vectorial Hamilton-Jacobi equations JF - SIAM J. Math. Anal. 29 (1998) 1481-1491 Y1 - 1998 A1 - Sandro Zagatti AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/3512 U1 - 752 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An existence result in a problem of the vectorial case of the calculus of variations Y1 - 1995 A1 - Arrigo Cellina A1 - Sandro Zagatti AB - SIAM J. Control Optim. 33 (1995) 960-970 PB - SIAM UR - http://hdl.handle.net/1963/3513 U1 - 751 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A version of Olech\\\'s lemma in a problem of the calculus of variations JF - SIAM J. Control Optim. 32 (1994) 1114-1127 Y1 - 1994 A1 - Arrigo Cellina A1 - Sandro Zagatti AB - 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. PB - SIAM UR - http://hdl.handle.net/1963/3514 U1 - 750 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - THES T1 - Some Problems in the Calculus of the Variations Y1 - 1992 A1 - Sandro Zagatti KW - Calculus of variations PB - SISSA UR - http://hdl.handle.net/1963/5428 U1 - 5260 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER -