An interpolating inequality for solutions of uniformly elliptic equations

We extend an inequality for harmonic functions, obtained by the same authors, 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 two papers by L. A. Caffarelli and by I. Blank and Z.Hao..