Partial regularity for $\mathscr{A}$-quasiconvex variational problems of linear growth

TL;DR

This paper establishes partial regularity results for minimizers of linear growth variational problems constrained by linear PDEs, extending previous results to more general operators and $x$-dependent integrands.

Contribution

It proves partial continuity of minimizers under strong $ ext{ extscr{A}}$-quasiconvexity for linear growth problems involving linear PDE operators of constant rank, including potential cases.

Findings

Minimizers are partially continuous under certain convexity conditions.

Results extend to $x$-dependent integrands and different linear PDE operators.

Analysis covers both direct and potential formulations of variational problems.

Abstract

We prove that minimizers of variational integrals $$ \mathcal E(v)=\int_\Omega f(v)\quad\text{for }v\in\mathcal M(\Omega)\text{ such that } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathscr{A}$-quasiconvex in a suitable sense. We consider linear growth problems, linear PDE operators $\mathscr{A}$ of constant rank, and variations of the form $v+\varphi$ with $\mathscr{A}$-free $\varphi\in \mathrm{C}_{\mathrm{c}}^\infty(\Omega)$. Our analysis also covers the ``potentials case'' $$ \mathcal F(u)=\int_\Omega f( \mathscr{B} u)\quad\text{for } u\in\mathscr D'(\Omega)\text{ such that }\mathscr B u\in \mathcal M(\Omega), $$ where $\mathscr{B}$ is a different linear pde operator of constant rank. Both our main results extend to $x$-dependent integrands.