Free boundary regularity for surfaces minimizing $\textrm {Area}(S)+c \textrm {Area}(\partial S)$

Abstract:

In ${{\mathbf {R}}^n}$, fix a hyperplane $Z$ and $a\;(k - 1)$-dimensional surface $F$ lying to one side of $Z$ with boundary in $Z$. We prove the existence of $S$ and $B$ minimizing $\operatorname {Area}(S) + c\operatorname {Area}(B)$ among all $k$-dimensional $S$ having boundary $F \cup B$, where $B$ is a free boundary constrained to lie in $Z$. We prove that except possibly on a set of Hausdorff dimension $k - 2$, $S$ is locally a ${C^{1,\alpha }}$ manifold with ${C^{1,\alpha }}$ boundary $B$ for $0 < \alpha < 1/2$. If $k = n - 1$, ${C^{1,\alpha }}$ is replaced by real analytic.