A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals

Abstract

In the present paper we show that the integral functional$I(y,u): = \int\limits_G {f(x,y(x),u(x))dx}$ is lower semicontinuous with respect to the joint convergence of y_k to y in measure and the weak convergence of u_k to u in L₁. The integrand f: G × ℝ^N × ℝ^m → ℝ, (x, z, p) → f(x, z, p) is assumed to be measurable in x for all (z,p), continuous in z for almost all x and all p, convex in p for all (x,z), and to satisfy the condition f(x,z,p)≧Φ(x) for all (x,z,p), where Φ is some L₁-function.

The crucial idea of our paper is contained in the following simple