A monotonicity formula for a classical free boundary problem

Abstract

We construct a monotonicity formula for the free boundary problem of the form $\hat{I}”u=\hat{I}¼$, where $\hat{I}¼$ is a Radon measure. It implies that the blow up limits of solutions are homogenous functions of degree one. The first formula is new even for classical Laplace operator. Our method of proof uses a careful application of the strong maximum principle.