Necessary and sufficient conditions in semicontinuity and convergence theorems with a~functional

M. A. Sychev

Sbornik. Mathematics, Tome 186 (1995) no. 6, pp. 847-878

Résumé

For the functional $$ {\mathfrak I}(u(x),\xi (x))=\int _\Omega L(x,u(x),\xi (x))\,dx $$ ($L(x,u,v)\colon{\mathbb R}^n\times{\mathbb R}^q\times{\mathbb R}^l\to{\mathbb R}$ satisfies the Caratheodory condition, and $L(x,u,v)\geqslant-\alpha(|u|+|v|)+\beta$, $\alpha>0$, $\beta\in{\mathbb R}$) it is proved that: 1) ${\mathfrak I}(u(x),\xi(x))$ is lower semicontinuous on a fixed pair $(u_0(x),\xi_0(x))$ of function $({\mathfrak I}(u_0(x),\xi_0(x))\infty)$ with respect to convergence of $u_k(x)$ to $u_0(x)$ in $L_1$ and weak convergence of $\xi_k(x)$ to $\xi_0(x)$ in $L_1$ if an only if for a.e. $x\in\Omega$ the function $L(x,u_0(x),v)$ is convex at the point $v=\xi_0(x)$; 2) strong convergence of $u_k(x)$ to $u_0(x)$ in $L_1$, weak convergence of $\xi_k(x)$ to $\xi _0(x)$ in $L_1$, and convergence of the values of the functional ${\mathfrak I}(u_k,\xi_k)$ to ${\mathfrak I}(u_0,\xi_0)\infty$ imply strong convergence of $\xi _k(x)$ to $\xi_0(x)$ if and only if for a.e. $x\in\Omega$ the function $L(x,u_0(x),v)$ is strictly convex at the point $v=\xi_0(x)$. Analogous results are obtained for problems with restrictions on the ranges of the functions $\xi_k(x)$ and in the gradient scalar case: $l=nq$, $\min\{n,q\}=1$, $\xi(x)=\nabla u(x)$.