A monotonicity formula for a classical free boundary problem
Aram Karakhanyan
TL;DR
We address a free boundary problem for local minimizers of $J(u)=\int_{\Omega}|\nabla u|^2+\lambda\,\chi_{\{u>0\}}$ by constructing a monotone density $K(r)=\frac{1}{|B_r|}\int_{B_r}\chi_{\{u>0\}}$. The approach combines a generalized maximum-principle framework with blow-up analysis, showing that $K(r)$ is nondecreasing and that blow-up limits are homogeneous of degree one; equality characterizes conical positivity sets. The analysis hinges on a function $w=\nabla u\cdot x-u$ satisfying $a_{ij}w_{ij}=b\cdot\nabla w$, and a case-by-case study (Cases 1–3) to force the vanishing of a key parameter $\ell^2$, with an Appendix establishing mean-curvature/regularity consequences at regular free-boundary points. Overall, the paper provides a robust monotonicity formula for a classical free boundary problem and clarifies the structure and rigidity of blow-up limits.
Abstract
We construct a monotonicity formula for the free boundary problem of the form $Δu=μ$, where $μ$ 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.