IUMJ

Title: On blow-ups and the classification of global solutions to parabolic free boundary problems

Authors: Kaj Nystrom

Issue: Volume 55 (2006), Issue 4, 1233-1290

Abstract:

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 $\Omega \subset \mathbb{R}^{n}$ is an unbounded $\delta$-Reifenberg flat domain, $\delta \in (0,\delta_{n})$, and if $\partial\Omega$ 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 $\partial\Omega$ and $\mathrm{d}\omega = k \mathrm{d}H^{n-1}$ where $\omega$ is the harmonic measure at infinity. If furthermore $\sup_{X\in\Omega} |\nabla u(X)| \leq 1$ and $k(Q) \geq 1$ for $H^{n-1}$ a.e. point $Q \in \partial\Omega$, 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 $\Omega = \{(x_0,x,t) \in \mathbf R \times \mathbb{R}^{n-1} \times \mathbb{R}: x_0 > \psi(x,t)\}$ where $\psi$ is a $\mathrm{Lip}(1,\tfrac12)$ function. Defining $\Omega^1 = \Omega$ and $\Omega^2 = \mathbb{R}^{n+1} \setminus \overline{\Omega}$, we let $\omega^i ( \hat{X^i}, \hat{t}^i , \cdot )$, for $i \in \{1,2\}$ and $(\hat{X}^i, \hat{t}^i ) \in \Omega^i$ be the caloric measure defined with respect to $\Omega^i$. Assuming that $\omega^i ( \hat{X}^i, \hat{t}^i , \cdot )$ is absolutely continuous with respect to an appropriate surface measure $\sigma$ for at least one $i \in \{1,2\}$, we study the implication of the condition $\log k^i ( \hat{X}^i, \hat{t}^i, \cdot ) \in VMO(\mathrm{d}\sigma)$ on the `free boundary' $\partial\Omega$. 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 $\partial\Omega$ 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.