Constraint maps with free boundaries: the Bernoulli case

Abstract

We study maps $u\in W^{1,2}(\Omega ;\overline{M})$ that minimize the Alt–Caffarelli energy functional

under the condition that the image $u(\Omega )$ is confined within $\overline{M}$. Here, $\Omega$ denotes a bounded domain in the ambient space ${\mathbb{R}}^n$ (with $n\ge 1$), and $M$ represents a smooth domain in the target space ${\mathbb{R}}^m$ (where $m\ge 2$). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, $\mathrm{int}(u^{-1}(\partial M))$, such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary. Our first significant contribution is an $\epsilon$-regularity theorem, founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small. Our subsequent key finding reveals that, whenever the complement of $M$ is uniformly convex and of class $C^3$, the maps minimizing the Alt–Caffarelli energy with a positive parameter $q$ exhibit Lipschitz continuity within a universally defined neighborhood of the noncoincidence set $u^{-1}(M)$. In particular, this Lipschitz continuity extends to the free boundary. A noteworthy consequence of our findings is the smoothness of flat free boundaries and of the resulting image maps.