Free boundary regularity close to initial state for parabolic obstacle problem
Henrik Shahgholian · Transactions of the American Mathematical Society · 2007
In this paper we study the behavior of the free boundary ∂ { u > ψ } \partial \{u>\psi \} , arising in the following complementary problem: ( H u ) ( u − ψ ) = 0 , u ≥ ψ ( x , t ) i n Q + , H u ≤ 0 , u ( x , t ) ≥ ψ ( x , t ) o n ∂ p Q + . \begin{gather*} (Hu)(u-\psi )=0,\qquad u\geq \psi (x,t) \quad \mathrm {in}\ Q^+, Hu \leq 0, u(x,t) \geq \psi (x,t) \quad \mathrm {on}\ \partial _p Q^+. \end{gather*} Here ∂ p \partial _p denotes the parabolic boundary, H H is a parabolic operator with certain properties, Q + Q^+ is the upper half of the unit cylinder in R n + 1 \textbf {R}^{n+1}