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Zapiski Nauchnykh Seminarov POMI, 2000, Volume 271, Pages 39–55
(Mi znsl1346)
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This article is cited in 14 scientific papers (total in 15 papers)
Boundary estimates for solutions to the parabolic free boundary problem
D. E. Apushkinskayaa, H. Shahgholianb, N. N. Ural'tsevaa a Saint-Petersburg State University
b Royal Institute of Technology
Abstract:
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 $u\in C^{1,1}_x\cap C^{0,1}_t$.
Received: 16.10.2000
Citation:
D. E. Apushkinskaya, H. Shahgholian, N. N. Ural'tseva, “Boundary estimates for solutions to the parabolic free boundary problem”, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Zap. Nauchn. Sem. POMI, 271, POMI, St. Petersburg, 2000, 39–55; J. Math. Sci. (N. Y.), 115:6 (2003), 2720–2730
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https://www.mathnet.ru/eng/znsl1346 https://www.mathnet.ru/eng/znsl/v271/p39
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