Geometry of Mean Value Sets for General Divergence Form Uniformly Elliptic Operators

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x₀ in the domain, there exists a nested family of sets {D_r(x₀)} where the average over any of those sets is related to the value of the function at x₀. Although it is known that the {D_r(x₀)} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that B_cr(x₀) ? D_r(x₀) ? B_Cr(x₀) for all r >?0 and x₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.