# Constraint maps with free boundaries: the Bernoulli case

Alessio Figalli, André Guerra, Sunghan Kim, Henrik Shahgholian

Alessio Figalli, André Guerra, Sunghan Kim, Henrik Shahgholian

**7/11/23**Published in : arXiv:2311.03006

In this manuscript, we delve into the study of maps 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, Ω denotes a bounded domain in the ambient space \mathbb{R}^n (with n≥1), and M represents a smooth domain in the target space \mathbb{R}^m (where m≥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 the validity of a ε-regularity theorem. This theorem is 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 energy. 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 non-coincidence 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