On the Rectifiability of the Free Boundary of the One Phase Stefan Problem
Marianne K. Korten · Communications in Analysis and Geometry · 2005
In one space dimension and for a given function u I (x) ∈ C ∞ 0 , (say such that u I (x) > 1 in some interval) the equation u t = ∆(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 0 ≤ u ∈ L 1 loc (IR n × (0, T )) 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 λ supported on the free boundary F = ∂{(x, t) : (u(x, t) -1) + > 0}.We prove that this measure is n rectifiable, i. e., F is λ-essentially the union of images of imbedded C 1 manifolds of dimension n in IR n × (0, T ), under a weak assumption on the spatial gradient of (u -1) + .