A theorem on lower semicontinuity of integral functionals

Abstract

A general lower semicontinuity theorem, in which not only mappings uM and PM but also the integrands fM depend on M, is proved for integrands f,fM under certain general hypotheses. These include thai f(x, u. P) is convex with respect to P and fM converge to f locally uniformly, but fM(x, u. P) are not required to be convex with respect to P and fM(x,·,·) do not even need to be lower semicontintious. Some more usable criteria for lower semicontinuity of integral funetionals are also given as corollaries of the main theorem.