On the pointwise jump condition at the free boundary in the one phase Stefan problem

Marianne K. Korten

Joint work with Donatella Danielli (Purdue University).

We consider locally integrable, nonnegative solutions in the sense of distributions of the one-phase Stefan problem

u_{t} = Δ(u − 1) + .

We recall that the measure λ = −div_{x,t}(∇(u − 1)_{+}, −(u − 1)_{+}) = (u − (u − 1)_{+})_{t} is supported on the free boundary F = ∂{(x, t) : (u − 1)_{+}(x, t) > 0}. Let C_{r}(x_{0}, t(x_{0})) = B_{r}(x_{0}) × (t_{0} − r, t_{0} + r). On any set E ⊂ IR^{n} × (0, T) such that for every (x_{0}, t_{0}) ∈ E ∩ F, lim inf_{r→0}{|∇(u − 1)_{+}(x, t), (x, t) ∈ C_{r}(x_{0}, t(x_{0}))} = b for some 0 < b = b(x_{0}) < ∞, λ restricted to E is carried by a countably n−rectifiable subset of F ∩ E.

Theorem Assume that in a such a set E ∩ F the n−density of λ is bounded away from 0. Them for H^{n} a. e. (x_{0}, t_{0}) ∈ E ∩ F

lim_{r→0} \frac{λ(C_{r})}{r^{n}} = (1 − u_{I})_{+}(x_{0})ν_{t}(x_{0}, t_{0}) = −L(x_{0}, t_{0})

where (ν_{x}, ν_{t}) in the outer unit normal to {(u − 1)_{+} > 0} at (x_{0}, t_{0}) ∈ F_{red}, L(x_{0}, t_{0}) the trace of (∇(u − 1)_{,}0) · ν(x_{0}, t_{0}), and u_{I} the (absolutely continuous part of the) initial data.

Steps in the proof consist in approaching the boundary measure in small enough cylinders centered at free boundary points by integrals along levet sets of (u − 1)_{+}. Under our hypothesis, the set {(u − 1)_{+} > 0} has finite perimeter in the set under consideration. Then we obtain a trace (attained in the sense of Gauss-Green’s theorem) for (∇(u − 1)_{,}0) · ν(x, t), at H^{n} a. e. point of the reduced free boundary. Equality of the densities of the disintegrations of the boundary measure λ yields the jump condtion.