The Mean Value Theorem and Basic Properties of the Obstacle Problem for Divergence Form Elliptic Operators

TL;DR

This paper provides a detailed proof of a mean value theorem for divergence form elliptic operators and explores fundamental properties of the obstacle problem, including nondegeneracy, for such operators.

Contribution

It offers a comprehensive proof of Caffarelli's mean value theorem and establishes basic properties of the obstacle problem for general elliptic divergence form operators.

Findings

Proof of Caffarelli's mean value theorem with detailed steps

Establishment of quadratic nondegeneracy property for obstacle problem

Basic properties of elliptic divergence form operators

Abstract

In 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by Caffarelli, and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegeneracy property.