Communications on Pure and Applied Analysis (CPAA)

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

Pages: 357 - 366, Volume 4, Issue 2, June 2005      doi:10.3934/cpaa.2005.4.357

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Donatella Danielli - Department of Mathematics, Purdue University, United States (email)
Marianne Korten - Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States (email)

Abstract: 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:  Stefan problem, mushy regions, discontinuous solutions, freebound-aries, jump condition, level sets, finite perimeter.
Mathematics Subject Classification:  35K65, 35R35

Received: March 2004;      Revised: November 2004;      Available Online: March 2005.