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_\Omega F(|\nabla u|^2)+\mbox{meas}({u>0}\cap \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-W\'eyl 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 $\R^3$.