Regularity for free boundary surfaces minimizing degenerate area functionals

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