Partial regularity and higher integrability for A-quasiconvex variational problems

Abstract

We prove that minimizers of variational problems on open sets $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ $$ \mbox{minimize}\quad \mathcal E(v)=\int_Ωf(v(x))\mathrm{d} x\quad\text{for } \mathscr{A} v=0, $$ are partially continuous provided that the integrands $f$ are strongly $\mathcal{A}$-quasiconvex in a suitable sense. We consider $p$-growth problems with $1<p<\infty$, linear constant rank PDE operators $\mathcal{A}$ on ${\mathbb{R}}^n$ between vector spaces $V$ and $W$, and Dirichlet boundary conditions, in the sense that admissible fields are of the form $v=v_0+\ddot{I}†$, with $\mathcal{A}$-free $\ddot{I}†\in C_c^{\infty }(\hat{I}\text{c◯},V)$. Our analysis also covers the ``potentials case'' $$ \mbox{minimize}\quad \mathcal F(u)=\int_Ωf(\mathscr{B} u(x))\mathrm{d} x\quad\text{for } u\in u_0+ C_c^\infty(Ω,U), $$ where $\mathcal{B}$ 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 $f(x,\mathcal{B}u(x))$.

38 pages. Some minor errors were corrected, and the proof of the partial regularity was shortened