## 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} = Δ(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.

Original language | English (US) |
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Pages (from-to) | 357-366 |

Number of pages | 10 |

Journal | Communications on Pure and Applied Analysis |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2005 |

Externally published | Yes |

## Keywords

- Discontinuous solutions
- Finite perimeter
- Free boundaries
- Jump condition
- Level sets
- Mushy regions
- Stefan problem

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics