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

In one space dimension and for a given function u_I(x) in C₀-infinity, (say such that u_I(x) \gt 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 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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Published 1 January 2005