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

Skipper, Jack; Wiedemann, Emil

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 107

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 $$ and the convex constraint is positive semi-definiteness of a matrix field.

Classification : 49J45, 35E10, 35Q35
Keywords: Convex sets, $$-quasiconvexity, $$-free, lower semi-continuity, Young measures, potentials, calculus of variations

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