A new monotonicity formula for quasilinear elliptic free boundary problems

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional $$J(u)={\int }_{\mathrm{\Omega }}F(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2)+meas(\{u>0\}\cap \mathrm{\Omega }),$$ 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 ${\mathbb{R}}^3$.