, Volume 115, Issue 6, pp 2720-2730

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

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Abstract

Let u and Ω solve the problem $$H(u) = X\Omega ,{\text{ }}u = |Du| = 0{\text{ }}in{\text{ }}Q_1^ + \backslash \Omega ,{\text{ }}u = 0{\text{ }}on{\text{ }}\Pi \cap Q_1 ,$$ where Ω is an open set in \(\begin{gathered} \mathbb{R}_ + ^{n + 1} = \{ (x,t):x \in \mathbb{R}^n ,t \in \mathbb{R}^1 ,x_1 >0\} ,n \geqslant 2,H = \Delta - \partial _t \\ \\ \end{gathered} \) is the heat operator, \(X\Omega \) denotes the characteristic function of Ω, \(Q_1 \) is the unit cylinder in ℝn+1, \(Q_1^ + = Q_1 \cap \mathbb{R}_ + ^{n + 1} ,\Pi = \{ (x,t):x1 = 0\} \) , and the first equation is satisfied in the sense of distributions. We obtain the optimal regularity of the function u, i.e., we show that \( \in C_x^{1,1} \cap C_t^{0,1} \) . Bibliography: 6 titles.