Regularity for solutions of the two-phase Stefan problem

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On periodic elliptic equations with gradient dependence

May 2008, 7(3): 591-600. doi: 10.3934/cpaa.2008.7.591

Regularity for solutions of the two-phase Stefan problem

Marianne Korten ^1, and Charles N. Moore ^1,

1.	Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States

Received March 2007 Revised October 2007 Published February 2008

Abstract
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We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.

Keywords: Stefan problem, degenerate parabolic equation..

Mathematics Subject Classification: Primary: 35K65; Secondary: 80A2.

Citation: Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591

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