Gradient regularity for strongly singular or degenerate elliptic and parabolic equations

TL;DR

This paper discusses recent progress in understanding the regularity of solutions to elliptic and parabolic equations with singular or degenerate structures, focusing on Besov and Sobolev regularity in various regimes.

Contribution

It provides new regularity results for weak solutions to classes of elliptic and parabolic equations with non-standard growth conditions, especially near singularities or degeneracies.

Findings

Regularity results in Besov and Sobolev spaces for elliptic equations.

Higher differentiability of solutions in parabolic equations.

Applicable to both subquadratic and superquadratic regimes.

Abstract

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and $p$-ellipticity conditions only outside a ball centered at the origin. In the elliptic setting, we describe Besov and Sobolev regularity results for suitable nonlinear functions of the gradient of the weak solutions, covering both the subquadratic ($1<p<2$) and superquadratic ($p\geq2$) regimes. Analogous results are obtained in the corresponding parabolic framework, where we address the higher spatial and temporal differentiability of the solutions under appropriate assumptions on the data.