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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.03006 |
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| _version_ | 1866914903008215040 |
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| author | Figalli, Alessio Guerra, André Kim, Sunghan Shahgholian, Henrik |
| author_facet | Figalli, Alessio Guerra, André Kim, Sunghan Shahgholian, Henrik |
| contents | In this manuscript, we delve into the study of maps $u\in W^{1,2}(Ω;\overline M)$ that minimize the Alt-Caffarelli energy functional $$ \int_Ω(|Du|^2 + q^2 χ_{u^{-1}(M)})\,dx, $$ under the condition that the image $u(Ω)$ is confined within $\overline M$. Here, $Ω$ 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, ${\rm int}(u^{-1}(\partial M))$, 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 $\varepsilon$-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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constraint maps with free boundaries: the Bernoulli case Figalli, Alessio Guerra, André Kim, Sunghan Shahgholian, Henrik Analysis of PDEs In this manuscript, we delve into the study of maps $u\in W^{1,2}(Ω;\overline M)$ that minimize the Alt-Caffarelli energy functional $$ \int_Ω(|Du|^2 + q^2 χ_{u^{-1}(M)})\,dx, $$ under the condition that the image $u(Ω)$ is confined within $\overline M$. Here, $Ω$ 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, ${\rm int}(u^{-1}(\partial M))$, 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 $\varepsilon$-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 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. |
| title | Constraint maps with free boundaries: the Bernoulli case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2311.03006 |