# Relaxation of p-growth integral functionals under space-dependent differential constraints

## TL;DR

This paper derives a representation formula for the relaxed form of integral energies involving space-dependent differential constraints and p-growth conditions, expanding the understanding of variational problems with variable coefficients.

## Contribution

It introduces a new relaxation formula for integral functionals with space-dependent constraints within the framework of -quasiconvexity, considering variable coefficients.

## Key findings

- Derived a representation formula for relaxed integral energies.
- Extended relaxation results to space-dependent differential constraints.
- Applicable to functionals with p-growth conditions.

## Abstract

A representation formula for the relaxation of integral energies $$(u,v)\mapsto\int_{\Omega} f(x,u(x),v(x))\,dx,$$ is obtained, where $f$ satisfies $p$-growth assumptions, $1<p<+\infty$, and the fields $v$ are subjected to space-dependent first order linear differential constraints in the framework of $\mathscr{A}$-quasiconvexity with variable coefficients.