TITLE

Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants

AUTHOR(S)
Cabezas-Rivas, Esther; Miquel, Vicente
PUB. DATE
January 2012
SOURCE
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p185
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
In a rotationally symmetric space $${{\overline M}}$$ around an axis $${\mathcal{A}}$$ (whose precise definition is satisfied by all real space forms), we consider a domain G limited by two equidistant hypersurfaces orthogonal to $${\mathcal{A}}$$. Let $${M \subset {\overline M}}$$ be a revolution hypersurface generated by a graph over $${\mathcal{A}}$$, with boundary in ? G and orthogonal to it. We study the evolution M of M under the volume-preserving mean curvature flow requiring that the boundary of M rests on ? G and stays orthogonal to it. We prove that: (a) the generating curve of M remains a graph; (b) the flow exists as long as M does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of M, the flow is defined for every $${t\in [0,\infty[}$$ and a sequence of hypersurfaces $${M_{t_n}}$$ converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.
ACCESSION #
67363487

 

Related Articles

  • Generalized Solution of a Kind of Nonparametric Curvature Evolution with Boundary Condition. Li Chen; Hui Liu // Acta Mathematica Sinica;Apr2006, Vol. 22 Issue 2, p455 

    The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on the reciprocal of the Gauss curvature.

  • A Boundary Value Problem for Hermitian Monogenic Functions. Blaya, Ricardo Abreu; Reyes, Juan Bory; Peña Peña, Dixan; Sommen, Frank // Boundary Value Problems;2008, p1 

    We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in ℝm, m = 2n. Necessary and sufficient conditions for the solvability of this problem are obtained.

  • Maximum Directed Cuts in Graphs with Degree Constraints. Xu, Baogang; Yu, Xingxing // Graphs & Combinatorics;Jul2012, Vol. 28 Issue 4, p563 

    The Max Cut problem is an NP-hard problem and has been studied extensively. Alon et al. (J Graph Theory 55:1-13, ) studied a directed version of the Max Cut problem and observed its connection to the Hall ratio of graphs. They proved, among others, that if an acyclic digraph has m edges and each...

  • Renormalized Energy Equidistribution and Local Charge Balance in 2D Coulomb Systems. Nodari, Simona Rota; Serfaty, Sylvia // IMRN: International Mathematics Research Notices;2015, Vol. 2015 Issue 11, p3035 

    We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization...

  • On asymptotic behavior for singularities of the powers of mean curvature flow. Sheng, Weimin; Wu, Chao // Chinese Annals of Mathematics;Feb2009, Vol. 30 Issue 1, p51 

    Let M n be a smooth, compact manifold without boundary, and F0: M n ? R n+1 a smooth immersion which is convex. The one-parameter families F(�, t): M n � [0, T) ? R n+1 of hypersurfaces M = F(�, t)( M n) satisfy an initial value problem d t/d F(�, t) = - H k(�, t)...

  • Applicability range of Stoney’s formula and modified formulas for a film/substrate bilayer. Yin Zhang; Ya-pu Zhao // Journal of Applied Physics;3/1/2006, Vol. 99 Issue 5, p053513 

    In addition to the layer thickness and effective Young’s modulus, the impact of the kinematic assumptions, interfacial condition, in-plane force, boundary conditions, and structure dimensions on the curvature of a film/substrate bilayer is examined. Different models for the analysis of...

  • Asymptotic behavior of least energy solutions for a singularly perturbed problem with nonlinear boundary condition. Abreu, Emerson; do Ó, João; Medeiros, Everaldo // Calculus of Variations & Partial Differential Equations;Jan2014, Vol. 49 Issue 1/2, p491 

    We consider the problem of finding a positive harmonic function $$u_\varepsilon $$ in a bounded domain $$\Omega \subset \mathbb R ^N (N\ge 3)$$ satisfying a nonlinear boundary condition of the form $$\varepsilon \partial _{\nu } u +u =|u|^{p-2}u,\,x\in \partial \Omega $$, where $$\varepsilon $$...

  • On a Free Boundary Problem for the Curvature Flow with Driving Force. Guo, Jong-Shenq; Matano, Hiroshi; Shimojo, Masahiko; Wu, Chang-Hong // Archive for Rational Mechanics & Analysis;Mar2016, Vol. 219 Issue 3, p1207 

    We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls...

  • On graphs in which the Hoffman bound for cocliques equals the Cvetcovich bound. Makhnev, A. A. // Doklady Mathematics;Jun2011, Vol. 83 Issue 3, p340 

    The article examines the regular graph wherein the Hoffman bound for cocliques is equivalent to the Cvetcovich bound. It notes that the graph employs various parameters and eigenvalues with Krein conditions, and addresses some problems associated with regular graphs without multiple edges or...

Share

Read the Article

Courtesy of VIRGINIA BEACH PUBLIC LIBRARY AND SYSTEM

Sign out of this library

Other Topics