Partial regularity and higher integrability for A-quasiconvex variational problems

Abstract

We prove that minimizers of variational problems on open sets $\Omega \subset \mathbb{R}^n$ $$ \mbox{minimize}\quad \mathcal E(v)=\int_\Omega f(v(x))\mathrm{d} x\quad\text{for } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathscr{A}$-quasiconvex in a suitable sense. We consider $p$-growth problems with $1<p<\infty$, linear constant rank PDE operators $\mathscr{A}$ on $\mathbb{R}^n$ between vector spaces $V$ and $W$, and Dirichlet boundary conditions, in the sense that admissible fields are of the form $v=v_0+\varphi$, with $\mathscr{A}$-free $\varphi\in C_c^\infty(\Omega,V)$. Our analysis also covers the ``potentials case'' $$ \mbox{minimize}\quad \mathcal F(u)=\int_\Omega f(\mathscr{B} u(x))\mathrm{d} x\quad\text{for } u\in u_0+ C_c^\infty(\Omega,U), $$ where $\mathscr{B}$ is another linear constant rank PDE operator on $\mathbb{R}^n$ between vector spaces $U,V$. We also prove appropriate higher integrability of minimizers for both types of problems. In addition, our approach covers non-autonomous integrands $f(x,v(x))$ or $f(x,\mathscr{B} u(x))$.