A free boundary problem inspired by a conjecture of De Giorgi
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2012
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| Online Access: | http://hdl.handle.net/1721.1/73368 |
| _version_ | 1826196546922741760 |
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| author | Kamburov, Nikola (Nikola Angelov) |
| author2 | David Jerison. |
| author_facet | David Jerison. Kamburov, Nikola (Nikola Angelov) |
| author_sort | Kamburov, Nikola (Nikola Angelov) |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. |
| first_indexed | 2024-09-23T10:29:06Z |
| format | Thesis |
| id | mit-1721.1/73368 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T10:29:06Z |
| publishDate | 2012 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/733682019-04-12T20:23:46Z A free boundary problem inspired by a conjecture of De Giorgi Kamburov, Nikola (Nikola Angelov) David Jerison. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. Cataloged from PDF version of thesis. Includes bibliographical references (p. 97-99). We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u) = f lVul2 + V(U), where V(u) is the characteristic function of the interval (-1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional J(u) = f lVul2 + W(u), where W(u) = (1 - u2 )2/2 is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of J that are bounded and monotone in one direction have levell sets that are hyperplanes, at least up to dimension 8. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension 9, whose level sets "follow" the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this thesis, we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase {lul < 1} will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only. by Nikola Kamburov. Ph.D. 2012-09-27T15:26:06Z 2012-09-27T15:26:06Z 2012 2012 Thesis http://hdl.handle.net/1721.1/73368 809647693 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 99 p. application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Kamburov, Nikola (Nikola Angelov) A free boundary problem inspired by a conjecture of De Giorgi |
| title | A free boundary problem inspired by a conjecture of De Giorgi |
| title_full | A free boundary problem inspired by a conjecture of De Giorgi |
| title_fullStr | A free boundary problem inspired by a conjecture of De Giorgi |
| title_full_unstemmed | A free boundary problem inspired by a conjecture of De Giorgi |
| title_short | A free boundary problem inspired by a conjecture of De Giorgi |
| title_sort | free boundary problem inspired by a conjecture of de giorgi |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/73368 |
| work_keys_str_mv | AT kamburovnikolanikolaangelov afreeboundaryprobleminspiredbyaconjectureofdegiorgi AT kamburovnikolanikolaangelov freeboundaryprobleminspiredbyaconjectureofdegiorgi |