Regularity theory for non-autonomous problems with a priori assumptions

Abstract

We study weak solutions and minimizers $u$ of the non-autonomous problems $\operatorname{div} A(x, Du)=0$ and $\min_v \int_\Omega F(x,Dv)\,dx$ with quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded, H\"older continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on $A$ or $F$ and the corresponding norm of $u$. We prove a Sobolev--Poincar\'e inequality, higher integrability and the H\"older continuity of $u$ and $Du$. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on $A$ or $F$ and assumptions on $u$ are known for the double phase energy $F(x, \xi)=|\xi|^p + a(x)|\xi|^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.