Regularity theory for non-autonomous problems with a priori assumptions

Abstract

We study weak solutions and minimizers $u$ of the non-autonomous problems $\mathrm{div}A(x,Du)=0$ and ${\min }_v{\int }_{\Omega }F(x,Dv)dx$ with quasi-isotropic $(p,q)$-growth. We consider the case that $u$ is bounded, Hölder 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--Poincaré inequality, higher integrability and the Hölder 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.

Calc. Var. Partial Differential Equations, to appear

References in corpus (2)

• Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces

Cited by in corpus (2)

• Regularity for multi-phase problems at nearly linear growth
• Local and global $C^{1,\beta }$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type