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Free boundary regularity close to initial state for parabolic obstacle problem

Authors:

Henrik Shahgholian

Institution:

Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

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,

is a parabolic operator with certain properties,

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.

Keywords:

Free boundary singular point obstacle problem regularity global solution blow-up initial state

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