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 $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminus\Omega)^a$ near $\partial\Omega$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,\gamma_0}$-surfaces that touch $\partial \Omega$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$.