A note on H\"older regularity of weak solutions to linear elliptic equations

TL;DR

This paper proves that weak solutions to certain linear elliptic equations with constant symmetric matrices are Hölder continuous, providing explicit regularity exponents and demonstrating sharpness of these results with a novel proof approach using a monotonicity formula.

Contribution

The paper establishes Hölder regularity for solutions with explicit exponents and introduces a new proof method via a monotonicity formula, extending results to variable coefficient cases.

Findings

Solutions are Hölder continuous with explicit exponent.

The regularity result is sharp for constant coefficient matrices.

A new proof technique using a monotonicity formula is developed.

Abstract

In this paper, we show that weak solutions of $$-\text{div} \mathbb{A}(x)\nabla u = 0 \qquad \text{where}\quad \mathbb{A}(x)= \mathbb{A}(x)^T \,\, \text{and} \,\, \lambda |\zeta|^2 \leq \langle \mathbb{A}(x)\zeta,\zeta\rangle \leq \Lambda |\zeta|^2,$$ and $\mathbb{A}(x) \equiv \mathbb{A}$ is a constant matrix are H\"older continuous $u \in C^{\alpha}_{\text{loc}}$ with $\alpha \geq \frac12 \left(-(n-2) + \sqrt{(n-2)^2 + \frac{4(n-1)\lambda}{\Lambda}} \right)$. This implies that the example constructed by Piccinini - Spagnolo is sharp in the class of constant matrices $\mathbb{A}(x) \equiv \mathbb{A}$. The proof of H\"older regularity does not go through a reduction of oscillation type argument and instead is achieved through a monotonicity formula. In the case of general matrices $\mathbb{A}(x)$, we obtain the same regularity under some additional hypothesis.