manuscripta mathematica

, Volume 27, Issue 1, pp 73-79

First online:

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

  • Goswin EisenAffiliated withMathematisches Institut der Universität Bonn

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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 yk to y in measure and the weak convergence of uk to u in L1. The integrand f: G × ℝN × ℝm → 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