A new monotonicity formula for quasilinear elliptic free boundary problems

Abstract

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_ΩF(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap Ω), \] where $F$ is a density function satisfying some structural conditions. The onus of proof lies with the careful analysis of the ghost function, the gradient part in the Helmholtz-Wéyl decomposition of a nonlinear flux that appears in the domain variation formula for $J(u)$. As an application we prove full regularity for a class of quasilinear Bernoulli type free boundary problems in ${\mathbb{R}}^3$.