Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary \partial \{u>\psi \}, arising in the following complementary problem: \begin{gather*} (Hu)(u-\psi)=0,\qquad u\geq \psi (x,t) \quad \hbox{in } Q^+, Hu \leq 0, u(x,t) \geq \psi (x,t) \quad \hbox{on } \partial_p Q^+. \end{gather*} Here \partia...

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Bibliographic Details
Published in:	Transactions of the American Mathematical Society 2008-04, Vol.360 (4), p.2077-2087
Main Author:	Shahgholian, Henrik
Format:	Article
Language:	eng
Subjects:	Boundary value problems Continuous functions Cylinders equations Exact sciences and technology Finance General mathematics General, history and biography Mathematical constants Mathematical functions Mathematical theorems Mathematics Maximum value Sciences and techniques of general use Viscosity
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Summary:	In this paper we study the behavior of the free boundary \partial \{u>\psi \}, arising in the following complementary problem: \begin{gather} (Hu)(u-\psi)=0,\qquad u\geq \psi (x,t) \quad \hbox{in } Q^+, Hu \leq 0, u(x,t) \geq \psi (x,t) \quad \hbox{on } \partial_p Q^+. \end{gather} Here \partial_p denotes the parabolic boundary, H is a parabolic operator with certain properties, Q^+ is the upper half of the unit cylinder in {\bf R}^{n+1}, and the equation is satisfied in the viscosity sense. The obstacle \psi is assumed to be continuous (with a certain smoothness at \{x_1=0, t=0\}), and coincides with the boundary data u(x,0)=\psi (x,0) at time zero. We also discuss applications in financial markets.
ISSN:	0002-9947 1088-6850 1088-6850