Boundary estimates for solutions to the parabolic free boundary problem

D. E. Apushkinskaya ; H. Shahgholian ; N. N. Ural'tseva

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Tome 271 (2000), pp. 39-55

Résumé

Let $u$ and $\varOmega$ (an open set in $\mathbb R^{n+1}_+=\{(x,t):x\in\mathbb R^n,\ t\in\mathbb R^1,\ x_1>0\}$, $n\geqslant2$) solve the following problem: $$ H(u)=\chi_{\varOmega}, \quad u=|Du|=0 \quad\text{in}\quad Q_1^+\setminus\varOmega, \quad u=0 \quad\text{on}\quad \Pi\cap Q_1, $$ where $H=\Delta-\partial_t$ is the heat operator, $\chi_{\varOmega}$ denotes the characteristic function of $\varOmega$, $Q_1$ is the unit cylinder in $\mathbb R^{n+1}$, $Q_1^+=Q_1\cap\mathbb R^{n+1}_+$, $\Pi=\{(x,t):x_1=0\}$, 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 $u\in C^{1,1}_x\cap C^{0,1}_t$.