Constraint maps with free boundaries: the Bernoulli case

We study maps u W^1, 2 (; M) that minimize the Alt–Caffarelli energy functional _ (|Du|^2 + q^2 ₔ^-₁ (M) ) \, dx under the condition that the image u () is confined within M. Here, denotes a bounded domain in the ambient space R^n (with n 1), and M represents a smooth domain in the target space R^m (where m 2). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, int (u^-1 (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 -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.