Interior regularity results for inhomogeneous anisotropic quasilinear equations

TL;DR

This paper establishes new interior regularity results for inhomogeneous anisotropic p-Laplace type equations, providing quantitative Sobolev regularity and weighted Hessian estimates under broad conditions.

Contribution

It introduces novel regularity results for anisotropic quasilinear equations, refining existing literature and extending understanding in non-Euclidean settings.

Findings

Quantitative Sobolev regularity for stress field a(∇u)

Weighted L^2 estimates for the Hessian of u

Results applicable to general anisotropic inhomogeneous equations

Abstract

We consider inhomogeneous $p$-Laplace type equations of the form $-\mathrm{div}\left(a(\nabla u)\right)=f$ in a possibly anisotropic setting. Under general assumptions on the source term $f$, we obtain quantitative Sobolev regularity results for the stress field $a(\nabla u)$ and weighted $L^2$ estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.