Regularity for free boundary surfaces minimizing degenerate area functionals

Abstract

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy ${\mathrm{Per}}_w(E)={\int }_{{\partial }^{\ast }E}w\mathrm{d}{\mathcal{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot ,{\mathbb{R}}^n\setminus \hat{I}\text{c◯}{)}^a$ near $\partial \hat{I}\text{c◯}$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,{\hat{I}}_0^3}$-surfaces that touch $\partial \hat{I}\text{c◯}$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathcal{H}}(\mathrm{Sing}(\partial E))\le n+a-(5+\sqrt{8})$.