Partial regularity and higher integrability for A-quasiconvex variational problems

We prove that minimizers of variational problems on open sets $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $$minimize\mathcal{E}(v)={\int }_{\mathrm{\Omega }}f(v(x))\mathrm{d}x\text{for }\mathcal{A}v=0,$$ are partially continuous provided that the integrands $f$ are strongly $\mathcal{A}$-quasiconvex in a suitable sense. We consider $p$-growth problems with $1<p<\mathrm{\infty }$, 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 $v=v_0+\phi$, with $\mathcal{A}$-free $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega },V)$. Our analysis also covers the ``potentials case’’ $$minimize\mathcal{F}(u)={\int }_{\mathrm{\Omega }}f(\mathcal{B}u(x))\mathrm{d}x\text{for }u\in u_0+C_c^{\mathrm{\infty }}(\mathrm{\Omega },U),$$ 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 $f(x,\mathcal{B}u(x))$.