arXiv 2207.034262022-07-07math.AP

Generalized Minimizing Movements for the varifold Canham-Helfrich flow

Katharina Brazda, Martin Kružík, Ulisse Stefanelli

英文摘要：

The gradient flow of the Canham-Helfrich functional is tackled via the Generalized Minimizing Movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered $C^{1,1}$ surfaces, we provide a Li-Yau-type estimate for the Canham-Helfrich energy and prove the conservation of multiplicity along the evolution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

两相膜演化的Cahn-Hilliard-Canham-Helfrich系统分析

Analysis of a Cahn-Hilliard-Canham-Helfrich system for the evolution of a two-phase membrane

Universität Regensburg（雷根斯堡大学）；Martin-Luther-Universität Halle-Wittenberg（马丁·路德·哈勒-维滕贝格大学）

本文推导了用于两相膜演化的新Cahn-Hilliard-Canham-Helfrich系统，并利用加权Lp空间中的拟线性抛物型演化方程理论证明了强局部时间解的存在性，从而表明系统是适定的。

2026-06-05math.AP

近圆柱管在Canham-Helfrich能量下的小变形及其在生物膜中的应用

Small deformations of a near cylindrical tube for the Canham-Helfrich Energy with applications to biological membranes

University of Warwick（华威大学）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；University of Sussex（萨塞克斯大学）

针对具有夹紧边界和面积约束的管状表面，提出一种近似Canham-Helfrich能量的二次能量，用于研究由点力或点约束引起的生物膜小变形，并推导了拉格朗日量和欧拉-拉格朗日方程，证明了适定性并给出数值示例。

2026-06-04math.AP

Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.

2020-02-10math.APmath.DGmath.OC

Canham-Helfrich模型中细胞膜的梯度流动力学

Gradient flow dynamics for cell membranes in the Canham-Helfrich model

Faculty of Mathematics, University of Vienna（维也纳大学数学学院）；Institute for Applied Mathematics, University of Bonn（波恩大学应用数学研究所）；Salzburg University（萨尔茨堡大学）

该研究针对Canham-Helfrich模型，通过多重性不等式证明初始能量低于阈值时，球体与轴对称环面的梯度流光滑解全局存在且收敛，为红细胞平衡恢复的数学描述提供了理论支撑。

2026-09-02math.APmath.DG

Asymptotic stability of local Helfrich minimizers

We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([12, 11]). The proof is based on a Lojasiewicz-Simon inequality.

2015-10-06math.AP

The Helfrich Boundary Value Problem

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower-semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Große-Brauckmann.

2018-09-28math.AP

更多相关论文收起更多论文

Existence of varifold minimizers for the multiphase Canham-Helfrich functional

We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally…

2020-03-05math-phmath.APmath.DG

Global minimizers for the doubly-constrained Helfrich energy: the axisymmetric case

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volume and having fixed surface area. By restricting attention to axisymmetric surfaces, we prove the existence of global minimizers.

2012-07-23math.APmath.DG

Global minimizers for axisymmetric multiphase membranes

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.

2012-04-30math-phmath.APmath.DG

Helfrich 泛函的 Li-Yau 不等式及其应用

Li-Yau inequalities for the Helfrich functional and applications

本文针对 Helfrich 泛函建立含自发曲率奇异体积项的一般 Li-Yau 不等式，并据此在球面 Canham-Helfrich 模型中证明低能量下光滑嵌入极小元的存在性。

2024-09-06math.APmath.DG

生物膜中由曲率驱动的图案形成：一种梯度流方法

Curvature-driven pattern formation in biomembranes: A gradient flow approach

本文提出了一种基于梯度流的相场模型，用于研究生物膜中由曲率驱动的图案形成，通过数值实验展示了不同参数对膜图案的影响。

2025-11-26math.AP