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

Abstract

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

45 pages