A note on Hölder regularity of weak solutions to linear elliptic equations

Abstract

In this paper, we show that weak solutions of $$-\text{div}\mathbb{A}(x)\nabla u=0\text{where}\mathbb{A}(x)=\mathbb{A}(x{)}^T\text{and}\lambda |\zeta {|}^2\le ⟨\mathbb{A}(x)\zeta ,\zeta ⟩\le \Lambda |\zeta {|}^2,$$ and $\mathbb{A}(x)\equiv \mathbb{A}$ is a constant matrix are Hölder continuous $u\in C_{\text{loc}}^{\alpha }$ with $\alpha \ge \frac{1}{2}\left(-(n-2)+\sqrt{(n-2{)}^2+\frac{4(n-1)\lambda }{\Lambda }}\right)$. 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 Hölder 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.