Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface

In this paper we show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution is smooth and satisfies the Hele-Shaw equation in the classical sense. A key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data. 0. Introduction. Consider a compact set K ⊂ Rn with smooth boundary ∂K. Suppose that a bounded domain Ω contains K and let Ω0 = Ω − K and Γ0 = ∂Ω (Figure 1). Note that ∂Ω0 = Γ0 ∪ ∂K. Let u0 be the harmonic function in Ω0 with u0 = f > 0 on K and zero on Γ0. Let u(x, t) solve the one phase Hele-Shaw problem:   −∆u = 0 in {u > 0} ∩ Q, ut − |∇u|2 = 0 on ∂{u > 0} ∩ Q, u(x, 0) = u0(x); u(x, t) = f for x ∈ ∂K. (HS) where Q = (Rn−K)× (0,∞). We refer to Γt(u) := ∂{u(·, t) > 0}−∂K as the free boundary of u at time t and to Ωt(u) := {u(·, t) > 0} as the positive phase. Note that if u is smooth up to the free boundary, then the free boundary moves with normal velocity V = ut/|∇u|, and hence the second equation in (HS) implies that V = |∇u|. The classical Hele-Shaw problem models an incompressible viscous fluid which occupies part of the space between two parallel, nearby plates. The short-time existence of classical solutions when Γ0 is C2+α was proved by Escher and Simonett [ES]. When n = 2, Elliot and Janovsky [EJ] showed the existence and uniqueness of weak solutions formulated by a parabolic variational inequality in H1(Q). For our investigation we use the notion of viscosity solutions introduced in [K1]. (See also Section 2.) In this paper we investigate general Lipschitz domain Ω0 in Rn with Lipschitz constant M less than a dimensional constant an. (In particular, a2 = 1.) Our main Manuscript received June 6, 2005; revised January 31, 2006. Research of the first author supported in part by NSF grant DMS-0503734; research of the second author supported in part by NSF grant DMS-0244991; and research of the third author supported in part by NSF grants DMS-0244991 and DMS-0401436. American Journal of Mathematics 129 (2007), 527–582. c © 2007 by The Johns Hopkins University Press. 527 528 S. CHOI, D. JERISON, AND I. KIM