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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free boundary regularity close to initial state for parabolic obstacle problem


Author: Henrik Shahgholian
Journal: Trans. Amer. Math. Soc. 360 (2008), 2077-2087
MSC (2000): Primary 35R35
Published electronically: November 19, 2007
MathSciNet review: 2366975
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Abstract: In this paper we study the behavior of the free boundary $ \partial \{u>\psi \}$, arising in the following complementary problem:

$\displaystyle (Hu)(u-\psi)=0,\qquad u\geq \psi (x,t) \quad \hbox{in } Q^+,$    
$\displaystyle Hu \leq 0,$    
$\displaystyle u(x,t) \geq \psi (x,t) \quad \hbox{on } \partial_p Q^+.$    

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.


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Additional Information

Henrik Shahgholian
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Email: henriksh@math.kth.se

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04292-4
PII: S 0002-9947(07)04292-4
Keywords: Free boundary, singular point, obstacle problem, regularity, global solution, blow-up, initial state.
Received by editor(s): January 7, 2005
Received by editor(s) in revised form: February 19, 2006
Published electronically: November 19, 2007
Additional Notes: This work was supported in part by the Swedish Research Council.
Article copyright: © Copyright 2007 American Mathematical Society