Differentiability of the Arrival Time
For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the crit...
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| Format: | Article |
| Language: | en_US |
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John Wiley & Sons
2016
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| Online Access: | http://hdl.handle.net/1721.1/105100 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| _version_ | 1826199343784263680 |
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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 | For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional 2 Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method, a~priori not even differentiable but only satisfying the equation in the viscosity sense . Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black-Scholes option pricing. From variations of the Sard and Łojasiewicz theorems, we relate differentiability to whether singularities all occur at only finitely many times for flows. |
| first_indexed | 2024-09-23T11:18:18Z |
| format | Article |
| id | mit-1721.1/105100 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:18:18Z |
| publishDate | 2016 |
| publisher | John Wiley & Sons |
| record_format | dspace |
| spelling | mit-1721.1/1051002022-09-27T18:37:21Z Differentiability of the Arrival Time Colding, Tobias Minicozzi, William Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional 2 Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method, a~priori not even differentiable but only satisfying the equation in the viscosity sense . Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black-Scholes option pricing. From variations of the Sard and Łojasiewicz theorems, we relate differentiability to whether singularities all occur at only finitely many times for flows. 2016-10-26T18:59:38Z 2016-10-26T18:59:38Z 2016-10 2015-02 Article http://purl.org/eprint/type/JournalArticle 00103640 http://hdl.handle.net/1721.1/105100 Colding, Tobias Holck, and William P. Minicozzi II. "Differentiability of the Arrival Time." Communications on Pure and Apploed Mathematics Volume 69, Issue 12 (December 2016), pp.2349–2363. https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 en_US http://dx.doi.org/10.1002/cpa.21635 Communications on Pure and Applied Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf John Wiley & Sons arXiv |
| spellingShingle | Colding, Tobias Minicozzi, William Differentiability of the Arrival Time |
| title | Differentiability of the Arrival Time |
| title_full | Differentiability of the Arrival Time |
| title_fullStr | Differentiability of the Arrival Time |
| title_full_unstemmed | Differentiability of the Arrival Time |
| title_short | Differentiability of the Arrival Time |
| title_sort | differentiability of the arrival time |
| url | http://hdl.handle.net/1721.1/105100 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| work_keys_str_mv | AT coldingtobias differentiabilityofthearrivaltime AT minicozziwilliam differentiabilityofthearrivaltime |