Lower semi-continuity for 𝒜-quasiconvex functionals under convex restrictions

Institute of Applied Analysis, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany.

^* Corresponding author: emil.wiedemann@uni-ulm.de

Received: 29 January 2021
Accepted: 30 November 2021

Abstract

We show weak lower semi-continuity of functionals assuming the new notion of a “convexly constrained” 𝒜-quasiconvex integrand. We assume 𝒜-quasiconvexity only for functions defined on a set K which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent “convexly constrained” 𝒜-free sequences. In a motivating example, the integrand is − det^1/d−1 and the convex constraint is positive semi-definiteness of a matrix field.