Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

Prosinski, Adam

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1605-1624.

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.

Classification : 35E99, 49J45
Mots clés : Closed A-p quasiconvexity, extended real-valued integrands, semi-continuity, Young measures, relaxation

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