The motion of a surface by its mean curvature

By K. Brakke. 2015

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In Computational Mechanics

Citations in the corpus

An algorithm for Mean Curvature Motion (2004, methodology): “Apart from the approaches of Brakke [19] and of Almgren, Taylor and Wang [2] and Luckhaus and Sturzenhecker [45], all other approaches, to our knowledge, are based on an inclusion principle which is the geometrical equivalent of the maximum principle for PDEs.”
Lower semicontinuity of the Willmore functional for currents (2009, methodology): “The converse is a consequence of the following more precise lemma recalling that quadratic height-excess decay almost everywhere implies quadratic tilt-excess decay almost everywhere under the assumption of square integrable weak mean curvature using the Cacciopoli-type estimate in [ Bra78 ] Theorem”
A sharp lower bound for the entropy of closed hypersurfaces up to dimension six (2014, background): “The second is the powerful regularity theory of Brakke [2] for such flows.”
Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “Later the evolution of submanifolds by their mean curvature has been studied by Brakke [Bra78] from the viewpoint of geometric measure theory.”
Regularity Theory for Mean Curvature Flow (2003, background): “As the author had stated in the Preface, this book was started as a set of informal notes on Brakke’s theory [1] about mean curvature flow.”
Generic mean curvature flow I; generic singularities (2009, methodology): “n take a sequence of parabolic dilations (x,t) → (cj x,c2 j t) with cj → ∞ to get MCF’s M j t = cj Mc−2 j t+t0 − x0 . Using Huisken’s monotonicity formula, [H3], and Brakke’s compactness theorem, [B], White [W4] and Ilmanen [I1] show that a subsequence of the Mj t ’s converges weakly to a limiting”
The volume preserving crystalline mean curvature flow of convex sets in R^N (2009, result): “Such a flow has been studied by many authors since the works of Brakke [11], Huisken [21], and Gage-Hamilton [20], and several results have been obtained in the last two decades on the subject.”
Embedded self-similar shrinkers of genus 0 (2014, background): “We ﬁrst recall a well-known result, which is an immediate consequence of Brakke’s local area bound (see [2] or [12, Prop.”
On short time existence for the planar network flow (2014, methodology): “After the pioneering work of Brakke [3], whose definition of moving varifolds includes the evolution of networks described here, and the fundamental results on curve shortening flow of embedded closed curves by Gage/Hamilton 1”
Singularity Profile in the Mean Curvature Flow (2009, neutral): “t inside the collapsing component, namely p0 ∈ Uk t −Nδ(Ut′) for all t ≤ −1 and for some δ > 0, where Nδ denotes the δ-neighborhood. Let Br(p1) be a ball in U′ −1. Let C = {q = tp0 +(1 − t)p : t ∈ [0,1], p ∈ Br(p1)} be a convex cone with vertex at p0. As in the proof of Lemma 5.1, let ¯tk = sup{s”
The singular set of mean curvature flow with generic singularities (2014, background): “These results were surveyed in [17]; see also [6,20], and [35] for regularity theory in MCF.”
Currents and flat chains associated to varifolds, with an application to mean curvature flow (2008, background): “Let I ⊂ R be an interval, typically either [0, ∞) or all of R. Recall that a Brakke flow t ∈ I 7→ V (t) of varifolds is called an integral Brakke flow provided V (t) is an integral varifold for almost all t ∈ I. (See [ Bra78 , §3] or [Ilm94, §6] for the definition of Brakke flow.)”
The round sphere minimizes entropy among closed self-shrinkers (2012, methodology): “It follows from Brakke’s theorem, [Br], that R has the least entropy of any self-shrinker and, in fact, there is a gap to the next lowest.”
Mean curvature flow with surgery of mean convex surfaces in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb (2013, neutral): “• γ̃r (t) = (cos(2π t), sin(2π t)) for r ∈ [1 2 , 1].”
Nonlocal Curvature Flows (2014, background): “On the other hand, Brakke [3] proposed a weak formulation for motion by mean curvature that resulted in deep regularity results but had the disadvantage of producing a lack of uniqueness.”