# A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators

## TL;DR

This paper establishes a characterization of mean value sets for elliptic divergence form operators as noncontact sets of obstacle problems involving the Green's function, linking mean value properties to obstacle problem solutions.

## Contribution

It provides a new equivalence between mean value sets and obstacle problem noncontact sets for elliptic divergence form operators.

## Key findings

- Mean value sets correspond exactly to obstacle problem noncontact sets.
- The connection is established via the Green's function of the elliptic operator.
- This characterization deepens understanding of mean value properties in elliptic PDEs.

## Abstract

We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form $Lu := \partial_i (a^{ij}(x) \partial_j u(x))$ if and only if it arises as the noncontact set of an obstacle problem involving the Green's function of the operator.