Communications in Analysis and Geometry

Volume 13 (2005)

Number 2

On the Rectifiability of the Free Boundary of the One Phase Stefan Problem

Pages: 363 – 378



Marianne K. Korten


In one space dimension and for a given function uI(x) in C0-infinity, (say such that uI(x) > 1 in some interval) the equation ut = delta(u - 1)+ can be thought of as describing the energy per unit volume in a Stefan-type problem, where the latent heat of the phase change is given by (1 - uI(x))+. Given a solution in the sense of distributions of this equation, (u - 1)+ is a subsolution to the heat equation. The "loss" with respect to a caloric function is accounted for by a Radon measure supported on the free boundary. We prove that this measure is n rectifiable, i. e., F is lambda-essentially the union of images of imbedded C1 manifolds of dimension n in Rn x (0, T), under a weak assumption on the spatial gradient of (u - 1)+.

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