Differentiability of the Arrival Time
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Colding_Differentiability of the.pdf
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171.29 KB
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Author(s)
Colding, Tobias
Minicozzi, William
Date Issued
October 2016
Journal
Communications on Pure and Applied Mathematics
Publisher
John Wiley & Sons
Citation
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.
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1002/cpa.21635
