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Semicontinuity problems in the calculus of variations. (English) Zbl 0537.49002

We are concerned with integral functionals of the form \(F(\Omega,u)=\int_{\Omega}f(x,u,Du)dx,\) where \(\Omega\) is a bounded open set of \({\mathbb{R}}^ n\) and \(u\in H^{1,p}(\Omega)\) for some \(p\geq 1\). Throughout the paper we assume that \(f:{\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}\) satisfies the following conditions: (a) \(0\leq f(x,s,\xi)\leq g(x,| s|,| \xi |),\) with g increasing with respect to \(| s|\) and \(| \xi |\), and locally summable in x. (b) f(x,s,\(\xi)\) is measurable with respect to x, upper semicontinuous with respect to \(\xi\), and continuous with respect to s uniformly as \(\xi\) varies on each bounded set of \({\mathbb{R}}^ n\). Two meaningful cases in which (b) is satisfied are the following: (i) \(f=f(x,\xi)\) is measurable in x and upper semicontinuous in \(\xi\) ; (ii) f is a Carathéodory function, i.e. measurable in x and continuous in \((x,\xi)\). We prove that the convexity of f with respect to \(\xi\) is a necessary condition for the s.l.s. (sequential lower semicontinuity) of F and we consider the following problem: if f is not convex with respect to \(\xi\), which is the greatest functional (s.l.s. in the weak topology o \(H^{1,p}(\Omega))\) which is less than or equal to \(F(\Omega\),\(\cdot)?\) We list some cases in which it is possible to characterize this functional as the integral \(\int_{\Omega}f^{**}(x,u,Du)dx,\) where \(f^{**}(x,s,\xi)\) is the greatest function (convex in \(\xi)\) which is less than or equal to f. Some unsolved problems remain when \(f^{**}\) is not a Carathéodory function. We apply these results to the relaxation of some variational problems. Relaxation means that, starting from a minimum problem for F which lacks a solution, a second minimum problem (the relaxed or generalized problem) is formulated involving a certain integral with the same infimum and whose optimal solutions are the limit points of the sequences minimizing the first problem. We study the relaxation for the Dirichlet and Neumann problems, for the obstacle problem and the relaxation for the convex set of functions with a prescribed bound on the Lipschitz constant.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49M20 Numerical methods of relaxation type
49Q20 Variational problems in a geometric measure-theoretic setting
90C25 Convex programming
49L99 Hamilton-Jacobi theories
54C08 Weak and generalized continuity
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