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

## TL;DR

This paper investigates the geometric properties of mean value sets associated with divergence form uniformly elliptic operators, extending classical mean value theorems and exploring their structure and applications.

## Contribution

It provides initial results on the geometry of mean value sets for general elliptic operators and discusses their potential applications.

## Key findings

- Mean value sets are nested and comparable to balls.
- Geometric and topological properties of these sets are characterized.
- Applications to related problems are demonstrated.

## 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.