Article

Journal of Mathematical Sciences

, Volume 115, Issue 6, pp 2720-2730

First online:

# Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

• D. E. ApushkinskayaAffiliated withSt.Petersburg State University
• , H. ShahgholianAffiliated withRoyal Institute of Technology
• , N. N. UraltsevaAffiliated withSt.Petersburg State University

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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 $$\begin{gathered} \mathbb{R}_ + ^{n + 1} = \{ (x,t):x \in \mathbb{R}^n ,t \in \mathbb{R}^1 ,x_1 >0\} ,n \geqslant 2,H = \Delta - \partial _t \hfill \\ \hfill \\ \end{gathered}$$ 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.