A two-phase free boundary with a logarithmic term

Abstract

We study minimizers of the energy functional ${\int }_{\Omega }|\nabla u{|}^2+2F(u)dx,$ where $F^{\prime }(u)\approx |u{|}^q\log u$ for some $-1<q<0$. We prove existence, optimal decay, and non-degeneracy of solutions, from free boundary points. Consequently, we derive the porosity property and an estimate on the Hausdorff dimension of the free boundary.