What question did this study set out to answer?

The aim is to advance the understanding of regularity for weak solutions in specific elliptic and parabolic equations.

February 28, 2026Open Access

Gradient regularity for strongly singular or degenerate elliptic and parabolic equations

Key Points

The aim is to advance the understanding of regularity for weak solutions in specific elliptic and parabolic equations.
Analyzed regularity theory for weak solutions to elliptic and parabolic equations.
Considered equations with standard $p$-growth and $p$-ellipticity conditions.
Explored subquadratic and superquadratic regimes for weak solutions.
Investigated Besov and Sobolev regularity results for gradients.
Established regularity for weak solutions outside a ball centered at the origin.
Provided results in both elliptic and parabolic frameworks.
Showed higher spatial and temporal differentiability under specific assumptions on data.

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 (p2) 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.

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