Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry
In this thesis we study the regularity of viscosity solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C¹,¹ submanifolds of spacetime with well understood behavior near singular times. We th...
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Published: |
Massachusetts Institute of Technology
2023
|
| Online Access: | https://hdl.handle.net/1721.1/152962 |
| _version_ | 1826207930116997120 |
|---|---|
| author | Hance, Jackson R. |
| author2 | Colding, Tobias H. |
| author_facet | Colding, Tobias H. Hance, Jackson R. |
| author_sort | Hance, Jackson R. |
| collection | MIT |
| description | In this thesis we study the regularity of viscosity solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C¹,¹ submanifolds of spacetime with well understood behavior near singular times. We then relate the derivatives of the solution of the level set flow to the solutions of certain evolution equations along fixed level sets. Finally we carry out this program to show that certain solutions with an axis of symmetry are in fact classical solutions of the level set problem. |
| first_indexed | 2024-09-23T13:57:12Z |
| format | Thesis |
| id | mit-1721.1/152962 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:57:12Z |
| publishDate | 2023 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1529622023-11-14T03:23:05Z Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry Hance, Jackson R. Colding, Tobias H. Massachusetts Institute of Technology. Department of Mathematics In this thesis we study the regularity of viscosity solutions to the level set equation for mean curvature flow. We describe a set of hypotheses under which we can prove that the level sets of these solutions are C¹,¹ submanifolds of spacetime with well understood behavior near singular times. We then relate the derivatives of the solution of the level set flow to the solutions of certain evolution equations along fixed level sets. Finally we carry out this program to show that certain solutions with an axis of symmetry are in fact classical solutions of the level set problem. Ph.D. 2023-11-13T19:57:34Z 2023-11-13T19:57:34Z 2023-09 2023-08-22T19:02:31.937Z Thesis https://hdl.handle.net/1721.1/152962 In Copyright - Educational Use Permitted Copyright retained by author(s) https://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Hance, Jackson R. Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title | Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title_full | Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title_fullStr | Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title_full_unstemmed | Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title_short | Regularity of the Level Set Equation for Mean Curvature Flow with an Axis of Symmetry |
| title_sort | regularity of the level set equation for mean curvature flow with an axis of symmetry |
| url | https://hdl.handle.net/1721.1/152962 |
| work_keys_str_mv | AT hancejacksonr regularityofthelevelsetequationformeancurvatureflowwithanaxisofsymmetry |