Nonconvexity and compact containment of mean value sets for second order uniformly elliptic operators in divergence form

Abstract

The mean value theorem for harmonic functions has historically been an important and powerful result. As such, a generalization of this theorem that was stated by Caffarelli in 1998 and later proved by Blank-Hao in 2015 is of immediate interest. However, in order to make more use of this new general mean value theorem, more information about the mean value sets that appear in the theorem is needed. We present here a few new results regarding properties of such mean value sets.