Constraint maps with free boundaries : the Bernoulli case

• In this manuscript, we delve into the study of maps that minimize the Alt–Caffarelli energy functional∫Ω(|Du|2 + q2χu−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 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 convexand 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 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 theresulting image maps.

Ämnesord

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

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