摘要 : 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 delta(n) > 0 such that if Omega subset of R-n is an unbounded delta-Reifenberg... 展开 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 delta(n) > 0 such that if Omega subset of R-n is an unbounded delta-Reifenberg flat domain, delta is an element of (0, delta(n)), and if partial derivative Omega 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 Delta u = 0 in Omega, u > 0 in Omega, u = 0 on partial derivative Omega and dw = kdH(n-1) where omega is the harmonic measure at infinity. If furthermore sup(x is an element of Omega)vertical bar del u(X)vertical bar <= 1 and k(Q) >= 1 for Hn-1 a.e. point Q is an element of partial derivative Omega, then in suitable coordinates, Omega = {(x, x(n)) : x(n) > 0} and u (x, x(n)) = x(n). 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 Omega = {(x(0), x, t) is an element of R x Rn-1 x R : x(0) > psi (x,t)} where psi is a Lip (1, 1/2) function.. Defining Omega(1) = Omega and Omega(2) = Rn+1\Omega, we let omega(i)(X-i, t(i),.), for i is an element of {1, 2} and (X-i, t(i)) is an element of Omega(i) be the caloric measure defined with respect to Omega(i). Assuming that omega(i) (X-i, t(i),.) is absolutely continuous with respect to an appropriate surface measure or for at least one i is an element of {1, 2}, we study the implication of the condition log k(i) (X-i, t(i) ,.) is an element of VMO(d sigma) on the 'free boundary' partial derivative Omega. 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 partial derivative Omega 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. 收起