Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

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.