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

摘要

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 x(0) in the domain, there exists a nested family of sets {D-r(x(0))} where the average over any of those sets is related to the value of the function at x(0). Although it is known that the {D-r(x(0))} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that B-cr(x(0)) subset of D-r(x(0)) subset of B-Cr(x(0)) for all r 0 and x(0) 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.