Partial regularity and higher integrability for A-quasiconvex variational problems (2412.10363v1)

Abstract: We prove that minimizers of variational problems $$ \mbox{minimize}\quad \mathcal E(v)=\int_\Omega f(x,v(x))\mathrm{d} x\quad\text{for } \mathscr{A} v=0, $$ 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 pde operators $\mathscr{A}$ of constant rank, and Dirichlet boundary conditions, in the sense that admissible fields are of the form $v=v_0+\varphi$, with $\mathscr{A}$-free $\varphi\in C_c^\infty(\Omega)$. Our analysis also covers the ``potentials case'' $$ \mbox{minimize}\quad \mathcal F(u)=\int_\Omega f(x,\mathscr{B} u(x))\mathrm{d} x\quad\text{for } u\in u_0+ C_c^\infty(\Omega), $$ where $\mathscr{B}$ is a different linear pde operator of constant rank. We also prove appropriate higher integrability of minimizers for both types of problems.