Level Set Method for Motion by Mean Curvature
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space...
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| Format: | Article |
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American Mathematical Society (AMS)
2018
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| Online Access: | http://hdl.handle.net/1721.1/115945 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| _version_ | 1811098257244291072 |
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| author | Colding, Tobias Minicozzi, William |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William |
| author_sort | Colding, Tobias |
| collection | MIT |
| description | Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties. |
| first_indexed | 2024-09-23T17:12:27Z |
| format | Article |
| id | mit-1721.1/115945 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T17:12:27Z |
| publishDate | 2018 |
| publisher | American Mathematical Society (AMS) |
| record_format | dspace |
| spelling | mit-1721.1/1159452022-10-03T11:08:48Z Level Set Method for Motion by Mean Curvature Colding, Tobias Minicozzi, William Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties. 2018-05-29T18:39:47Z 2018-05-29T18:39:47Z 2016-11 2018-05-17T15:57:27Z Article http://purl.org/eprint/type/JournalArticle 0002-9920 1088-9477 http://hdl.handle.net/1721.1/115945 Colding, Tobias Holck and William P. Minicozzi. “Level Set Method for Motion by Mean Curvature.” Notices of the American Mathematical Society 63, 10 (November 2016): 1148–1153 © 2016 American Mathematical Society https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 http://dx.doi.org/10.1090/NOTI1439 Notices of the American Mathematical Society Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Mathematical Society (AMS) American Mathematical Society |
| spellingShingle | Colding, Tobias Minicozzi, William Level Set Method for Motion by Mean Curvature |
| title | Level Set Method for Motion by Mean Curvature |
| title_full | Level Set Method for Motion by Mean Curvature |
| title_fullStr | Level Set Method for Motion by Mean Curvature |
| title_full_unstemmed | Level Set Method for Motion by Mean Curvature |
| title_short | Level Set Method for Motion by Mean Curvature |
| title_sort | level set method for motion by mean curvature |
| url | http://hdl.handle.net/1721.1/115945 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| work_keys_str_mv | AT coldingtobias levelsetmethodformotionbymeancurvature AT minicozziwilliam levelsetmethodformotionbymeancurvature |