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 \, \, ||² A (x), ||², and A (x) A is a constant matrix are H\"older continuous u C^₋₎₂ with 12 (- (n-2) + (n-2) ² + 4 (n-1) {}). 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.