Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners
The goal of this note is to continue the investigation started in Part One of the structure of "blown up" sets of the form $mathcal{P}imes mathbb{R}$ and $mathcal{N}imes mathbb{R}$ when $mathcal{P}, mathcal{N} subset mathbb{R}^{2}$ and $mathcal{P}$ (or $mathcal{N}$) minimizes an appro...
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| Format: | Article |
| Language: | English |
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Texas State University
2008-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2008/160/abstr.html |
| _version_ | 1811219442558828544 |
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| author | Kirk Lancaster Thalia Jeffres |
| author_facet | Kirk Lancaster Thalia Jeffres |
| author_sort | Kirk Lancaster |
| collection | DOAJ |
| description | The goal of this note is to continue the investigation started in Part One of the structure of "blown up" sets of the form $mathcal{P}imes mathbb{R}$ and $mathcal{N}imes mathbb{R}$ when $mathcal{P}, mathcal{N} subset mathbb{R}^{2}$ and $mathcal{P}$ (or $mathcal{N}$) minimizes an appropriate functional and the domain has a nonconvex corner. Sets like $mathcal{P}imes mathbb{R}$ can be the limits of the blow ups of subgraphs of solutions of capillary surface or other prescribed mean curvature problems, for example. Danzhu Shi recently proved that in a wedge domain $Omega$ whose boundary has a nonconvex corner at a point $O$ and assuming the correctness of the Concus-Finn Conjecture for contact angles $0$ and $pi$, a capillary surface in positive gravity in $Omegaimesmathbb{R}$ must be discontinuous under certain conditions. As an application, we extend the conclusion of Shi's Theorem to the case where the prescribed mean curvature is zero without any assumption about the Concus-Finn Conjecture. |
| first_indexed | 2024-04-12T07:25:51Z |
| format | Article |
| id | doaj.art-3d407aebd74a42c8a5771f0b938b4ffa |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-12T07:25:51Z |
| publishDate | 2008-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-3d407aebd74a42c8a5771f0b938b4ffa2022-12-22T03:42:11ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912008-12-012008160,125Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex cornersKirk LancasterThalia JeffresThe goal of this note is to continue the investigation started in Part One of the structure of "blown up" sets of the form $mathcal{P}imes mathbb{R}$ and $mathcal{N}imes mathbb{R}$ when $mathcal{P}, mathcal{N} subset mathbb{R}^{2}$ and $mathcal{P}$ (or $mathcal{N}$) minimizes an appropriate functional and the domain has a nonconvex corner. Sets like $mathcal{P}imes mathbb{R}$ can be the limits of the blow ups of subgraphs of solutions of capillary surface or other prescribed mean curvature problems, for example. Danzhu Shi recently proved that in a wedge domain $Omega$ whose boundary has a nonconvex corner at a point $O$ and assuming the correctness of the Concus-Finn Conjecture for contact angles $0$ and $pi$, a capillary surface in positive gravity in $Omegaimesmathbb{R}$ must be discontinuous under certain conditions. As an application, we extend the conclusion of Shi's Theorem to the case where the prescribed mean curvature is zero without any assumption about the Concus-Finn Conjecture.http://ejde.math.txstate.edu/Volumes/2008/160/abstr.htmlBlow-up setscapillary surfaceConcus-Finn conjecture |
| spellingShingle | Kirk Lancaster Thalia Jeffres Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners Electronic Journal of Differential Equations Blow-up sets capillary surface Concus-Finn conjecture |
| title | Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners |
| title_full | Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners |
| title_fullStr | Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners |
| title_full_unstemmed | Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners |
| title_short | Vertical blow ups of capillary surfaces in $R^3$, Part 2: Nonconvex corners |
| title_sort | vertical blow ups of capillary surfaces in r 3 part 2 nonconvex corners |
| topic | Blow-up sets capillary surface Concus-Finn conjecture |
| url | http://ejde.math.txstate.edu/Volumes/2008/160/abstr.html |
| work_keys_str_mv | AT kirklancaster verticalblowupsofcapillarysurfacesinr3part2nonconvexcorners AT thaliajeffres verticalblowupsofcapillarysurfacesinr3part2nonconvexcorners |