Constraint maps with free boundaries: the Bernoulli case

We study maps u\in W^{1,2}(\Omega;\overline M) that minimize the Alt–Caffarelli energy functional \int_{\Omega} (|Du|^{2} + q^{2} \chi_{u^{-1}(M)})\,dx 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\geq 1 ), and M represents a smooth domain in the target space \mathbb{R}^{m} (where m\geq 2 ). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, {\rm 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 \varepsilon -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.