Higher Hölder regularity for degenerate elliptic PDEs with data in Morrey spaces

TL;DR

The paper addresses the problem of obtaining sharp local $C^{1,\alpha}$ regularity for weak solutions of degenerate elliptic PDEs of $p$-Laplacian type with data in Morrey spaces, specifically $f\in L^{1,\lambda}(\Omega)$ with $n-1<\lambda<n$. The authors leverage the Fefferman--Phong inequality as a weighted Poincaré-type estimate and employ a comparison with solutions to the corresponding homogeneous equation along with Campanato–Meyer characterizations to control the nonlinear right-hand side and obtain gradient Hölder continuity. They establish explicit, sharp formulas for the Hölder exponent: for $\frac{2n}{\lambda+1}<p\le 2$, $\alpha = \min\bigl(\gamma,\lambda+1-\frac{2n}{p}\bigr)$, and for $2\le p< n$, $\alpha = \min\bigl(\gamma,\frac{\lambda+1-n}{p-1}\bigr)$, where $\gamma$ is the gradient regularity of the homogeneous problem; these results extend the linear theory ($p=2$) to nonlinear degenerate settings. The paper also extends these results to a broad class of quasilinear equations with analogous structural and Morrey-space RHS assumptions, providing a unified framework for higher regularity under modest data assumptions. This work advances the understanding of regularity in degenerate PDEs with minimal right-hand side integrability and Morrey-type decay, with potential implications for nonlinear PDE theory and related applications.

Abstract

We establish sharp local $C^{1,α}$-regularity for weak solutions to degenerate elliptic equations of $p$-Laplacian type with data in Morrey spaces. The proof relies on the Fefferman-Phong inequality and standard tools from regularity theory for nonlinear PDEs.

Theorems & Definitions (19)

• Definition 2.1
• Remark 2.1
• Proposition 2.1
• Definition 2.2
• Lemma 2.1
• Theorem 2.1
• Corollary 2.1
• Theorem 2.2
• Definition 3.1
• Remark 3.1