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

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.