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

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