manuscripta mathematica

, Volume 27, Issue 1, pp 73–79

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

  • Goswin Eisen
Article

DOI: 10.1007/BF01297738

Cite this article as:
Eisen, G. Manuscripta Math (1979) 27: 73. doi:10.1007/BF01297738

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 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

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Goswin Eisen
    • 1
  1. 1.Mathematisches Institut der Universität BonnBonn 1