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June  2005, 4(2): 357-366. doi: 10.3934/cpaa.2005.4.357

On the pointwise jump condition at the free boundary in the 1-phase Stefan problem

Donatella Danielli 1, and  Marianne Korten 2,

1. 

Department of Mathematics, Purdue University, United States
2. 

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

Received  March 2004 Revised  November 2004 Published  March 2005

In this paper we obtain the jump (or Rankine-Hugoniot) condition at the interphase for solutions in the sense of distributions to the one phase Stefan problem $u_t= \Delta (u-1)_+.$ We do this by approximating the free boundary with level sets, and using methods from the theory of bounded variation functions. We show that the spatial component of the normal derivative of the solution has a trace at the free boundary that is picked up in a natural sense. The jump condition is then obtained from the equality of the $n$-density of two different disintegrations of the free boundary measure. This is done under an additional condition on the $n$-density of this measure. In the last section we show that this condition is optimal, in the sense that its satisfaction depends on the geometry of the initial data.
Keywords: finite perimeter., mushy regions, Stefan problem, level sets, jump condition, freebound-aries, discontinuous solutions.
Mathematics Subject Classification: 35K65, 35R35.
Citation: Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357
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