Constraint maps with free boundaries: the Bernoulli case

We study maps u∈ W1,2(Ω; M) that minimize the Alt–Caffarelli energy functional ∫Ω (|Du|² + q² χ_{u⁻¹(M)})\,dx under the condition that the image u(Ω) is confined within M . Here, Ω denotes a bounded domain in the ambient space Rⁿ (with n≥ 1 ), and M represents a smooth domain in the target space Rᵐ (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 ε -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 C³ , 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.