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

In this paper, we show that weak solutions of $$-div A(x)∇ u = 0 where A(x)= A(x)^T \,\, and \,\, λ |ζ|^2 ≤ A(x)ζ,ζ ≤ Λ |ζ|^2,$$ and $A(x) ≡ A$ is a constant matrix are H\"older continuous $u ∈ Cαloc$ with $α ≥ 12 (-(n-2) + √(n-2)^2 + 4(n-1)λ/Λ )$. This implies that the example constructed by Piccinini - Spagnolo is sharp in the class of constant matrices $A(x) ≡ A$. 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 $A(x)$, we obtain the same regularity under some additional hypothesis.