The free-boundary Brakke flow

Abstract

Abstract. We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for freeboundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White [23] to the free-boundary setting. Using Ilmanen's elliptic regularization procedure [10], we prove existence of free-boundary Brakke flows.

Mean Convex Mean Curvature Flow with Free Boundary

“…the space-time track of the free boundary level set flow. Moreover, it can be checked (see Theorem 1.6 and Remark 1.7 below) that (1.4) M h fH n @K t g t!0 is a free boundary Brakke flow as defined in [10] (see also Section 5). The pair .M; K/ is called a mean-convex free boundary flow.…”

“…Likewise, the number n is sharp for general barriers, since the surface can pop when it hits @D, cf. [10]. For example, if D h x B 4 n B 1 & R ng1 is an annulus and K h x…”

“…Given X i h .x i ; t i / 3 X h .x; t/ in the support of the flow and i 3 I, we consider the blowup sequence .M i ; K i / that is obtained from .M; K/ by translating X i to the origin and parabolically rescaling by i . It is always possible to find a convergent subsequence where M i converges in the sense of (free boundary) Brakke flows [10,25] and K i converges in the Hausdorff sense. Any subsequential limit .M H ; K H / is called a limit flow at X.…”

“…If X i h X, we call it a tangent flow at X. Tangent flows are always backwardly self-similar [10,20].…”

“…In the smooth setting, mean curvature flow with free-boundary has been studied in important work by Stahl [41,42] and Buckland [7]. In the singular setting, in analogy to the flow of closed hypersurfaces, the free-boundary flow can be described as a level-set flow [13] or as a Brakke-type flow [10]. We prove here the existence of a unique mean-convex free boundary flow for all time and establish sharp results on the size and structure of singularities for this flow.…”