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We continue our study of the free boundary regularity in the thin one-phase problem, and show that C2,α free boundaries are smooth.

In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.

A version of a famous and important result due to Alt-Caffarelli, relevant to the analysis of elliptic free boundary problems, states that there exists δn > 0 such that if Ω ⊂ ℝn is an unbounded δ-Reifenberg flat domain, δ ϵ (0, δn), and if ∂Ω satisfies an Ahlfors condition, then the following is true. Assume that there exist functions u (Green function with pole at infinity) and k (the Poisson kernel) such that Δu = 0 in Ω, u > 0 in Ω, u = 0 on ∂Ω and dω = kdHn–1 where ω is the harmonic measure at infinity. If furthermore supXϵΩ |∇u(X)| ≤ 1 and k(Q) ≥ 1 for Hn–1 a.e. point Q ϵ ∂Ω, then in suitable coordinates, Ω = {(x,xn) : xn > 0} and u(x,xn) = xn. This result is crucial in recent work on the analysis of elliptic free boundary problems beyond the continuous threshold by Kenig and Toro. In this paper we consider the corresponding parabolic problems in the setting of time-varying domains Ω = {(x0,x,t) ∈ R × ℝn–1 × ℝ : x0 > Ψ(x,t)} where Ψ is a Lip(1,½) function. Defining Ω1 = Ω and Ω2 = ℝn+1 \\ Ω̄, we let ωi(X̂i,t̂i,·), for i ∈ {1,2} and (X̂i,t̂i) ∈ Ωi be the caloric measure defined with respect to Ωi. Assuming that ωi(X̂i,t̂i,·) is absolutely continuous with respect to an appropriate surface measure σ for at least one i ∈ {1,2}, we study the implication of the condition log ki(X̂i,t̂i,·) ∈ VMO(dσ) on the 'free boundary' ∂Ω. We show that this information on the Poisson kernel(s) can be explored in a delicate blow-up argument and that results on the regularity of ∂Ω can be deduced from classification theorems for global solutions to parabolic free boundary problems appearing in the limit. In fact, we prove a number of such classification theorems and, in particular, we prove weaker parabolic analogues of the result of Alt-Caffarelli.

We consider the one-dimensional one phase Stefan problems with temperature or thermal flux boundary conditions. Exact and closed-form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions, and so are the similarity solutions. In general, only numerical methods allow one to completely solve such nonlinear problems. In this paper, the quasi-similarity solutions are defined and, for each problem, an analytical approximation of the interfacial position is given as function of the time-dependent boundary datum.

In this paper, the following one-phase Stefan problem is considered: u1=uxx+f(u,t) in Q(T), u(x,0)=u0(x), 0⩽x⩽l0, \\[ u x(0,t) = 0,0 < t T,\\]u(l(t),t)=0, 0 < t ⩽T, l'(t)=-ux(l(t),t),l(0)=l0, where Ql(T)=(x,t)|00. It is proved that when the solution is blow-up in a finite time s(uo), and u0(x) is not a constant, then the free boundary will not be blow-up and the blow-up set is contained in the interval [0,l0). Moreover, when f(u,t)=u1+μ for some μ>0, every blow-up point is isolated.

We consider the supercooled one-phase Stefan problem with convective boundary condition at the fixed face. We analyse the relation between the heat transfer coefficient and the possibility of continuing the solution for arbitrarily large time intervals.