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# 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

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n2.a4

#### Author

#### Abstract

In one space dimension and for a given function *u*_{I}(*x*) in *C*_{0}-infinity, (say such that *u*_{I}(*x*) > 1 in some interval) the equation *u*_{t} = 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 - *u*_{I}(*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 *C*^{1} manifolds of dimension *n* in *R*^{n} x (0, *T*), under a weak assumption on the spatial gradient of (*u* - 1)_{+}.