An interpolating inequality for solutions of uniformly elliptic equations

Abstract

We extend an inequality for harmonic functions, obtained in [15,17], to the case of solutions of uniformly elliptic equations in divergence form, with merely measurable coefficients. The inequality for harmonic functions turned out to be a crucial ingredient in the study of the stability of the radial symmetry for Alexandrov's Soap Bubble Theorem and Serrin's problem. The proof of our inequality is based on a mean value property for elliptic operators stated and proved in [8] and [7].