On mean curvature flow with forcing

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. We prove that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become smooth in finite time. Lastly, for a model problem, we will discuss the flow's exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed in [24] to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.

Volume preserving mean curvature flow for star-shaped sets

“…Roughly speaking this property amounts to the boundary of the set being Lipschitz with respect to the spherical coordinate given by B ρ (0). [KK18] shows, by moving planes argument, that this property is preserved in the flow with volume-dependent forcing, which includes (1.3). In particular this property implies (b) for Ω δ t , as well as an equi-continuity over time, yielding the first part of (a).…”

“…Compared to the original flow (1.1), (1.3) holds an advantage that λ δ (t) only depends on |Ω t |, thus it can be handled with little information on the regularity of Γ t , which makes it easier to handle with viscosity solutions theory. The existence and uniqueness for viscosity solutions of (1.3) were proved in [KK18]. The following is summary of the main results in Theorem 3.1 & 4.1, Corollary 4.2 and Theorem 5.1.…”

“…Motivated by recent results [MSS16] and [KK18], our strategy is to approximate (1.1) by the following flow as δ → 0:…”

“…While it is suspected that star-shapedness is preserved in the evolution, it remains open to be proved. In [KK18] we instead considered a stronger version of star-shapeness, i.e. the property ρ-reflection given in Section 2.…”

Head and Tail Speeds of Mean Curvature Flow with Forcing

“…In this subsection, we prove a variant version of the comparison principle regarding a pseudo viscosity subsolution and a pseudo viscosity supersolution. The idea here is partially motivated by [5] and [17]. Then by the assumptions, we have t 0 ∈ (α, β) and Z(x, t 0 ) = 1 > 0 = W(x, t 0 ) for some x / ∈ ∂Ω(0, R; ν).…”