# Closed $\mathcal{A}$-$p$ Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

## TL;DR

This paper explores the relationship between lower semi-continuity of integral functionals and a new form of quasiconvexity called closed -p quasiconvexity, providing conditions for relaxation and extending classical results in calculus of variations.

## Contribution

It introduces closed -p quasiconvexity and establishes its equivalence to lower semi-continuity under certain conditions, extending the theory to extended real-valued integrands.

## Key findings

- Identifies the lower semi-continuous envelope for extended real-valued integrands.
- Shows the equivalence of closed -p quasiconvexity and lower semi-continuity with additional assumptions.
- Discusses the impact of dropping continuity assumptions on quasiconvexity and lower semi-continuity.

## Abstract

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 $\mathcal{A}$-$p$ 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 $\mathcal{A}$.