Geometry of mean value sets for general divergence form uniformly elliptic operators

Abstract

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 $B_{cr}(x_0)\subset D_r(x_0)\subset B_{Cr}(x_0)$ 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.

arXiv comment15 pages, 1 figure