Accepted Paper

Inserted: 8 aug 2024
Last Updated: 19 aug 2024

Journal: J. Eur. Math. Soc. (JEMS)
Year: 2024

Abstract:

In this manuscript, we delve into the study of maps \( u \in W^{1,2}(\Omega;M) \) that minimize the Alt–Caffarelli energy functional \[ \int_{\Omega} \left(
Du
^2 + q^2 \chi_{u^{-1}(M)} \right) \, dx, \] under the condition that the image \( u(\Omega) \) is confined within \( M \). 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, \( \operatorname{int}(u^{-1}(\partial M)) \), 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 an \( \varepsilon \)-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.

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.

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• Constraint-maps-with-free-boundaries-the-Bernoulli-case.pdf