Differentiability of the Arrival Time
Differentiability of the Arrival Time
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到达时间的可微分
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
W. Minicozzi
中科院分区：
文献类型：
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作者：
T. Colding;W. Minicozzi
对于单调前进的波前，到达时间是波前到达给定点的时间。我们证明了它是处处二次可微的，且具有一致有界的二阶导数。在远离方程退化的临界点处，它是光滑的。我们还证明了临界集具有有限余维2 Hausdorff测度.对于单调推进的波前，到达时间等价于水平集方法，先验甚至不可微，仅满足粘性意义下的方程。利用它是二次可微的，我们可以确定在临界点的海森，我们证明了它满足经典意义下的方程。到达时间具有博弈论解释。对于线性热方程，有一个与布莱克-斯科尔斯期权定价有关的博弈论解释。从Sard和Jasojasiewicz定理的变体中，我们将可微性与流的奇点是否只出现2000次联系起来。© 2016 Wiley Periodicals，Inc.
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.