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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 $v$ aresubjected to space-dependent first order linear differential constraints in theframework of $\mathscr{A}$-quasiconvexity with variable coefficients.

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