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Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants
In a rotationally symmetric space around an axis (whose precise definition is satisfied by all real space forms), we consider a domain G limited by two equidistant hypersurfaces orthogonal to . Let be a revolution hypersurface generated by a graph over , with boundary in ∂ G and orthogonal to it. We...
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| Published in: | Calculus of variations and partial differential equations 2012-01, Vol.43 (1-2), p.185-210 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c414t-3c67bd953caefc1ee9269c4511dc863ac8249921387cdd0932ae4bd4c68b3faf3 |
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| cites | cdi_FETCH-LOGICAL-c414t-3c67bd953caefc1ee9269c4511dc863ac8249921387cdd0932ae4bd4c68b3faf3 |
| container_end_page | 210 |
| container_issue | 1-2 |
| container_start_page | 185 |
| container_title | Calculus of variations and partial differential equations |
| container_volume | 43 |
| creator | Cabezas-Rivas, Esther Miquel, Vicente |
| description | In a rotationally symmetric space
around an axis
(whose precise definition is satisfied by all real space forms), we consider a domain
G
limited by two equidistant hypersurfaces orthogonal to
. Let
be a revolution hypersurface generated by a graph over
, with boundary in ∂
G
and orthogonal to it. We study the evolution
M
t
of
M
under the volume-preserving mean curvature flow requiring that the boundary of
M
t
rests on ∂
G
and stays orthogonal to it. We prove that: (a) the generating curve of
M
t
remains a graph; (b) the flow exists as long as
M
t
does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of
M
, the flow is defined for every
and a sequence of hypersurfaces
converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions. |
| doi_str_mv | 10.1007/s00526-011-0408-9 |
| format | article |
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around an axis
(whose precise definition is satisfied by all real space forms), we consider a domain
G
limited by two equidistant hypersurfaces orthogonal to
. Let
be a revolution hypersurface generated by a graph over
, with boundary in ∂
G
and orthogonal to it. We study the evolution
M
t
of
M
under the volume-preserving mean curvature flow requiring that the boundary of
M
t
rests on ∂
G
and stays orthogonal to it. We prove that: (a) the generating curve of
M
t
remains a graph; (b) the flow exists as long as
M
t
does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of
M
, the flow is defined for every
and a sequence of hypersurfaces
converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.</description><identifier>ISSN: 0944-2669</identifier><identifier>EISSN: 1432-0835</identifier><identifier>DOI: 10.1007/s00526-011-0408-9</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Analysis ; Boundaries ; Boundary conditions ; Calculus of variations ; Calculus of Variations and Optimal Control; Optimization ; Control ; Curvature ; Evolution ; Graphs ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Rest ; Systems Theory ; Theoretical ; Touch</subject><ispartof>Calculus of variations and partial differential equations, 2012-01, Vol.43 (1-2), p.185-210</ispartof><rights>Springer-Verlag 2011</rights><rights>Springer-Verlag 2012</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c414t-3c67bd953caefc1ee9269c4511dc863ac8249921387cdd0932ae4bd4c68b3faf3</citedby><cites>FETCH-LOGICAL-c414t-3c67bd953caefc1ee9269c4511dc863ac8249921387cdd0932ae4bd4c68b3faf3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27923,27924</link.rule.ids></links><search><creatorcontrib>Cabezas-Rivas, Esther</creatorcontrib><creatorcontrib>Miquel, Vicente</creatorcontrib><title>Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants</title><title>Calculus of variations and partial differential equations</title><addtitle>Calc. Var</addtitle><description>In a rotationally symmetric space
around an axis
(whose precise definition is satisfied by all real space forms), we consider a domain
G
limited by two equidistant hypersurfaces orthogonal to
. Let
be a revolution hypersurface generated by a graph over
, with boundary in ∂
G
and orthogonal to it. We study the evolution
M
t
of
M
under the volume-preserving mean curvature flow requiring that the boundary of
M
t
rests on ∂
G
and stays orthogonal to it. We prove that: (a) the generating curve of
M
t
remains a graph; (b) the flow exists as long as
M
t
does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of
M
, the flow is defined for every
and a sequence of hypersurfaces
converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.</description><subject>Analysis</subject><subject>Boundaries</subject><subject>Boundary conditions</subject><subject>Calculus of variations</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Control</subject><subject>Curvature</subject><subject>Evolution</subject><subject>Graphs</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Rest</subject><subject>Systems Theory</subject><subject>Theoretical</subject><subject>Touch</subject><issn>0944-2669</issn><issn>1432-0835</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp1kE1LxDAURYMoOI7-AHfBlZvqS5OmzVLELxhwoy7chDR9HTu0zUySzjD_3g4VBMHV3Zx7uRxCLhncMID8NgBkqUyAsQQEFIk6IjMmeJpAwbNjMgMlRJJKqU7JWQgrAJYVqZiRzw_XDh3StceAftv0S9qh6akd_NbEwSOtW7ejrqYetyMaG9fTr_0afRh8bSwGWmLcIfY07hzFzdBUTYimj-GcnNSmDXjxk3Py_vjwdv-cLF6fXu7vFokVTMSEW5mXlcq4NVhbhqhSqazIGKtsIbmx40-lUsaL3FYVKJ4aFGUlrCxKXpuaz8n1tLv2bjNgiLprgsW2NT26IWgmc8Y5yAJG9OoPunKD78d3WkEmpcjHH3PCJsh6F4LHWq990xm_1wz0QbaeZOtRtj7I1mrspFMnjGy_RP87_H_pG3hXg7w</recordid><startdate>20120101</startdate><enddate>20120101</enddate><creator>Cabezas-Rivas, Esther</creator><creator>Miquel, Vicente</creator><general>Springer-Verlag</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope><scope>7TB</scope><scope>7U5</scope><scope>8FD</scope><scope>FR3</scope><scope>H8D</scope><scope>KR7</scope><scope>L7M</scope></search><sort><creationdate>20120101</creationdate><title>Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants</title><author>Cabezas-Rivas, Esther ; Miquel, Vicente</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c414t-3c67bd953caefc1ee9269c4511dc863ac8249921387cdd0932ae4bd4c68b3faf3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Analysis</topic><topic>Boundaries</topic><topic>Boundary conditions</topic><topic>Calculus of variations</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Control</topic><topic>Curvature</topic><topic>Evolution</topic><topic>Graphs</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Rest</topic><topic>Systems Theory</topic><topic>Theoretical</topic><topic>Touch</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cabezas-Rivas, Esther</creatorcontrib><creatorcontrib>Miquel, Vicente</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Aerospace Database</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Calculus of variations and partial differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cabezas-Rivas, Esther</au><au>Miquel, Vicente</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants</atitle><jtitle>Calculus of variations and partial differential equations</jtitle><stitle>Calc. Var</stitle><date>2012-01-01</date><risdate>2012</risdate><volume>43</volume><issue>1-2</issue><spage>185</spage><epage>210</epage><pages>185-210</pages><issn>0944-2669</issn><eissn>1432-0835</eissn><abstract>In a rotationally symmetric space
around an axis
(whose precise definition is satisfied by all real space forms), we consider a domain
G
limited by two equidistant hypersurfaces orthogonal to
. Let
be a revolution hypersurface generated by a graph over
, with boundary in ∂
G
and orthogonal to it. We study the evolution
M
t
of
M
under the volume-preserving mean curvature flow requiring that the boundary of
M
t
rests on ∂
G
and stays orthogonal to it. We prove that: (a) the generating curve of
M
t
remains a graph; (b) the flow exists as long as
M
t
does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of
M
, the flow is defined for every
and a sequence of hypersurfaces
converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s00526-011-0408-9</doi><tpages>26</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0944-2669 |
| ispartof | Calculus of variations and partial differential equations, 2012-01, Vol.43 (1-2), p.185-210 |
| issn | 0944-2669 1432-0835 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_1671330680 |
| source | Springer Nature |
| subjects | Analysis Boundaries Boundary conditions Calculus of variations Calculus of Variations and Optimal Control Optimization Control Curvature Evolution Graphs Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Rest Systems Theory Theoretical Touch |
| title | Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants |
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