A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder
In this paper we consider a mean curvature flow \(V=H+A\) in a high dimensional cylinder \(\Omega\times \R\), where, \(A\) is a constant, \(\Omega\) is a bounded domain in \(\R^n\), and, for a hypersurface \(y=u(x,t)\) over \(\Omega\), \(V\) and \(H\) denote its normal velocity and mean curvature, r...
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| description | In this paper we consider a mean curvature flow \(V=H+A\) in a high dimensional cylinder \(\Omega\times \R\), where, \(A\) is a constant, \(\Omega\) is a bounded domain in \(\R^n\), and, for a hypersurface \(y=u(x,t)\) over \(\Omega\), \(V\) and \(H\) denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary \(\partial \Omega \times \R\) with prescribed angle \(\theta(x)\). Under certain assumptions such as \(\Omega\) is strictly convex and \(\|\cos\theta\|_{C^2}\) is small, or \(\Omega\) is not necessarily convex but \(|A|\) is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when \(I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0\) (resp. \(=0\), \( |
| doi_str_mv | 10.48550/arxiv.2210.16475 |
| format | Article |
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Assume the hypersurface contacts the cylinder boundary \(\partial \Omega \times \R\) with prescribed angle \(\theta(x)\). Under certain assumptions such as \(\Omega\) is strictly convex and \(\|\cos\theta\|_{C^2}\) is small, or \(\Omega\) is not necessarily convex but \(|A|\) is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when \(I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0\) (resp. \(=0\), \(<0\)), the solution \(u\) converges as \(t\to \infty\) to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.2210.16475</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Asymptotic properties ; Boundary value problems ; Contact angle ; Curvature ; Cylinders ; Hyperspaces ; Mathematics - Differential Geometry</subject><ispartof>arXiv.org, 2024-02</ispartof><rights>2024. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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Assume the hypersurface contacts the cylinder boundary \(\partial \Omega \times \R\) with prescribed angle \(\theta(x)\). Under certain assumptions such as \(\Omega\) is strictly convex and \(\|\cos\theta\|_{C^2}\) is small, or \(\Omega\) is not necessarily convex but \(|A|\) is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when \(I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0\) (resp. \(=0\), \(<0\)), the solution \(u\) converges as \(t\to \infty\) to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).</description><subject>Asymptotic properties</subject><subject>Boundary value problems</subject><subject>Contact angle</subject><subject>Curvature</subject><subject>Cylinders</subject><subject>Hyperspaces</subject><subject>Mathematics - Differential Geometry</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GOX</sourceid><recordid>eNotj11rwjAYhcNgMHH-gF0tsOu6fL1NcindnAPHBvO-pG2qkZq6pNX57-10VwceDofzIPRAyVQoAPJswq87TBkbAE2FhBs0YpzTRAnG7tAkxi0hhKWSAfAR-p7hD2s8zvpwMF0fLJ437REfXbfBX8HGMrjCVjhrfWfKDs_8urERO48NXrj1Br-4nfXRtd40ODs1zlc23KPb2jTRTv5zjFbz11W2SJafb-_ZbJkYDZAIzithZU0EgVKVohKgmFZcF4XQqoSaUisNSF5qOrC0qqGCuuZcmEFHMz5Gj9fZi3G-D25nwin_M88v5kPj6drYh_ant7HLt20fhqsxZ5ITpRkRkp8Bdt1aJQ</recordid><startdate>20240216</startdate><enddate>20240216</enddate><creator>Gao, Zhenghuan</creator><creator>Bendong Lou</creator><creator>Xu, Jinju</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20240216</creationdate><title>A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder</title><author>Gao, Zhenghuan ; Bendong Lou ; Xu, Jinju</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a955-433d4e7f0405c8c4d45829839bb498c5f11e7a573c919bb6df5d5ff334a331923</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Asymptotic properties</topic><topic>Boundary value problems</topic><topic>Contact angle</topic><topic>Curvature</topic><topic>Cylinders</topic><topic>Hyperspaces</topic><topic>Mathematics - Differential Geometry</topic><toplevel>online_resources</toplevel><creatorcontrib>Gao, Zhenghuan</creatorcontrib><creatorcontrib>Bendong Lou</creatorcontrib><creatorcontrib>Xu, Jinju</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gao, Zhenghuan</au><au>Bendong Lou</au><au>Xu, Jinju</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder</atitle><jtitle>arXiv.org</jtitle><date>2024-02-16</date><risdate>2024</risdate><eissn>2331-8422</eissn><abstract>In this paper we consider a mean curvature flow \(V=H+A\) in a high dimensional cylinder \(\Omega\times \R\), where, \(A\) is a constant, \(\Omega\) is a bounded domain in \(\R^n\), and, for a hypersurface \(y=u(x,t)\) over \(\Omega\), \(V\) and \(H\) denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary \(\partial \Omega \times \R\) with prescribed angle \(\theta(x)\). Under certain assumptions such as \(\Omega\) is strictly convex and \(\|\cos\theta\|_{C^2}\) is small, or \(\Omega\) is not necessarily convex but \(|A|\) is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when \(I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0\) (resp. \(=0\), \(<0\)), the solution \(u\) converges as \(t\to \infty\) to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.2210.16475</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Asymptotic properties Boundary value problems Contact angle Curvature Cylinders Hyperspaces Mathematics - Differential Geometry |
| title | A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder |
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