Worn stones with flat sides all time regularity of the interface

In this paper, we study the deformation of the 2-dimensional convex surfaces in R 3 whose speed at a point on the surface is proportional to α-power of positive part of Gauss Curvature. First, for 1 2 < α 1, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with C 1,1 -estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 < t < T 0 under certain necessary regularity and non-degeneracy initial conditions, where T 0 is the vanishing time of the flat side.

show abstract

“…Then Lemma 5.3 tells us the linearized equation (5.3) is in the same class of operators considered in Lemma 5.2 of [10] and [9].…”

Section: Regularity Theorymentioning

confidence: 92%

“…In [10], the authors have obtained the C 2,γ s regularity of h on the box. Following the proof Theorem 6.10 of [10], we will have the following theorem.…”

Section: Regularity Theorymentioning

confidence: 97%

α-Gauss Curvature flows with flat sides

Kim

Lee

Rhee

2013

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…It follows from the work of B. Andrews in [2] that, in this case, all surfaces become instantly of class C 1,1 and remain so up to a time when they become strictly convex and therefore smooth, before they contract to a point. Also, it follows from the works of the first author with Hamilton [7] and Lee [8] that C 1,1…”

Section: Introductionmentioning

confidence: 99%

C 1, α regularity of solutions to parabolic Monge-Ampére equations

Daskalopoulos¹,

Savin²

2012

ajm

View full text Add to dashboard Cite

Abstract. We study interior C 1,α regularity of viscosity solutions of the parabolic Monge-Ampére equationwith exponent p > 0 and with coefficients b which are bounded and measurable.We show that when p is less than the critical power 1 n−2 then solutions become instantly C 1,α in the interior. Also, we prove the same result for any power p > 0 at those points where either the solution separates from the initial data, or where the initial data is C 1,β .

show abstract

“…Many physical phenomena such as flame propagation [1], crystal growth [2,3], and wear of rolling stones [4] can be described by curvature dependent front evolutions. For a given planar curve 0 ( ), the curvature dependent normal evolution ( , ) is defined by the equation…”

Section: Introductionmentioning

confidence: 99%