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Differentiability of the Arrival Time
Author(s) -
Colding Tobias Holck,
Minicozzi William P.
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21635
Subject(s) - differentiable function , mathematics , almost everywhere , monotonic function , interpretation (philosophy) , mathematical analysis , bounded function , programming language , computer science
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 Wiley Periodicals, Inc.