On Compactness of Minimizing Sequences Subject to a Linear Differential Constraint

Abstract

For $\Omega \subset {\mathbb{R}}^N$ open, we consider integral functionals of the form

defined on the subspace of $L^p$ consisting of those vector fields $u$ which satisfy the system $\mathcal{A}u=0$ on $\Omega$ in the sense of distributions. Here, $\mathcal{A}$ may be any linear differential operator of first order with constant coefficients satisfying Murat's condition of constant rank. The main results provide sharp conditions for the compactness of minimizing sequences with respect to the strong topology in $L^p$. Although our results hold for bounded domains as well, our main focus is on domains with infinite measure, especially exterior domains.