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# Communications in Analysis and Geometry

## Volume 23 (2015)

### Number 1

### The mean value theorem and basic properties of the obstacle problem for divergence form elliptic operators

Pages: 129 – 158

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n1.a4

#### Authors

#### Abstract

In 1963, Littman *et al.* 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.