抛物 Allen-Cahn 方程 Dirichlet 问题解的收敛性——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1768-1790.

抛物 Allen-Cahn 方程 Dirichlet 问题解的收敛性——献给李工宝教授 70 寿辰

王常健¹, 郑高峰²^,^*()

¹信阳师范大学数学与统计学院河南信阳 464000
²华中师范大学数学与统计学学院, 非线性分析教育部重点实验室, 数学物理湖北省重点实验室武汉 430079

收稿日期:2025-01-06 修回日期:2025-04-25 出版日期:2025-12-26 发布日期:2025-11-18
通讯作者: 郑高峰 E-mail:gfzheng@mail.ccnu.edu.cn
基金资助:
国家自然科学基金(12271195);国家自然科学基金(12401144);河南省自然科学基金(242300421695)

Convergence of Solutions to Dirichlet Problem of Parabolic Allen-Cahn Equation

Changjian Wang¹, Gaofeng Zheng²^,^*()

¹School of Mathematics and Statistics, Xinyang Normal University, Henan Xinyang 464000
²School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis and Applications (Ministry of Education), Hubei Provincial Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan 430079

Received:2025-01-06 Revised:2025-04-25 Online:2025-12-26 Published:2025-11-18
Contact: Gaofeng Zheng E-mail:gfzheng@mail.ccnu.edu.cn
Supported by:
NSFC(12271195);NSFC(12401144);NSF of Henan Province(242300421695)

摘要/Abstract

摘要：

关于 Allen-Cahn 方程收敛性问题的研究, 目前主要集中于 Neumann 边值问题, 然而对相关的其他边值问题却鲜有研究. 该文主要探讨当参数趋于 0 时, 由抛物 Allen-Cahn 方程 Dirichlet 边值问题诱导的极限 varifold 是 Brakke 意义下的平均曲率流.

关键词: 抛物 Allen-Cahn 方程, Dirichlet 问题, Brakke 平均曲率流

Abstract:

The studies on the convergence of Allen-Cahn equation mainly focus on Neumann boundary value problems, but there are few researches on other related boundary value problems. This paper mainly explores the limit varifold induced by the Dirichlet boundary value problem of the parabolic Allen-Cahn equation when the parameters tend to 0, which is Brakke's mean curvature flow.

Key words: parabolic Allen-Cahn equation, Dirichlet problem, Brakke's mean curvature flow

中图分类号:

O175.23

王常健, 郑高峰. 抛物 Allen-Cahn 方程 Dirichlet 问题解的收敛性——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1768-1790.

Changjian Wang, Gaofeng Zheng. Convergence of Solutions to Dirichlet Problem of Parabolic Allen-Cahn Equation[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1768-1790.

参考文献 31

[1]	Abels H, Moser M. Convergence of the Allen-Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to $90^{\circ}$. SIAM J Math Anal, 2022, 54(1): 114-172 doi: 10.1137/21M1424925
[2]	Allard W K. On the first variation of a varifold. Ann of Math, 1972, 95(2): 417-491 doi: 10.2307/1970868
[3]	Allen S M, Cahn J W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Met, 1979, 27(6): 1085-1095 doi: 10.1016/0001-6160(79)90196-2
[4]	Ambrosto L, Gigli N, Savare G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Second Edition, Lectures in Mathematics ETH Zurich. Basel: Birkhäuser Verlag, 2008
[5]	Bethuel F, Orlandi G, Smets D. Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. Ann of Math, 2006, 163(2): 37-163 doi: 10.4007/annals
[6]	Brakke K A. The Motion of A Surface by its Mean Curvature. Mathematical Notes,Vol 20, Princeton: Princeton University Press, 1978
[7]	Bretin E, Perrier V. Phase field method for mean curvature flow with boundary constraints. ESAIM Math Model Numer Anal, 2012, 46(6): 1509-1526 doi: 10.1051/m2an/2012014
[8]	Bronsard L, Kohn R V. Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J Differential Equations, 1991, 90: 211-237 doi: 10.1016/0022-0396(91)90147-2
[9]	Bellettini G. Lecture Notes on Mean Curvature Flow:Barriers and Singular Perturbations. Pisa: Scuola Normale Superiore, 2013
[10]	Chen X. Generation and propagation of interfaces for reaction-diffusion equations. J Differential Equations, 1992, 96(1): 116-141 doi: 10.1016/0022-0396(92)90146-E
[11]	Chen X, Elliott C M, Gardiner A, Zhao J. Convergence of numerical solutions to the Allen-Cahn equation. Appl Anal, 1998, 69: 47-56
[12]	Chen Y, Giga Y, Goto S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J Differential Geom, 1991, 33: 749-786
[13]	De Lellis C. Rectifiable Sets, Density and Tangent Measures. Zurich: European Mathematical Society, 2008
[14]	Evans L, Spruck J. Motion of level sets by mean curvature, I. J Differential Geom, 1991, 33: 635-681
[15]	Evans L C, Soner H M, Souganidis P E. Phase transitions and generalized motion by mean curvature. Comm Pure Appl Math, 1992, 45: 1097-1123 doi: 10.1002/cpa.v45:9
[16]	Giga Y, Onoue F, Takasao K. A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions. Differential Integral Equations, 2021, 34: 21-126
[17]	Han Q. Nonlinear Elliptic Equations of the Second Order. Volume 171. Providence: Amer Math Soc, 2016
[18]	Ilmanen T. Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J Differential Geom, 1993, 38: 417-461
[19]	Jiang G C, Wang C J, Zheng G F. Convergence of solutions of some Allen-Cahn equations to Brakke's mean curvature flow. Acta Appl Math, 2020, 167(1): 149-169 doi: 10.1007/s10440-019-00272-2
[20]	Katsoulakis M, Kossioris G T, Reitich F. Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions. J Geom Anal, 1995, 5: 255-279 doi: 10.1007/BF02921677
[21]	Kagaya T. Convergence of the Allen-Cahn equation with Neumann boundary condition on non-convex domains. Math Ann, 2019, 373: 1485-1528 doi: 10.1007/s00208-018-1720-x
[22]	Kim L, Tonegawa Y. On the mean curvature flow of grain boundaries. Ann Inst Fourier (Grenoble), 2017, 67(1): 43-142 doi: 10.5802/aif.3077
[23]	Mizuno M, Tonegawa Y. Convergence of the Allen-Cahn equation with Neumann boundary conditions. SIAM J Math Anal, 2015, 47(3): 1906-1932 doi: 10.1137/140987808
[24]	Modica L. Gradient theory of phase transitions and minimal interface criteria. Arch Rational Mech Anal, 1987, 98: 123-142 doi: 10.1007/BF00251230
[25]	Owen N, Rubinstein J, Sternberg P. Minimizers and gradient flows for singularly perturbed bi-stable potentials with a dirichlet condition. Proc Roy Soc London A, 1990, 429: 505-532
[26]	Qi Y W, Zheng G F. Convergence of solutions of the weighted Allen-Cahn equations to Brakke type flow. Calc Var Partial Differential Equations, 2018, 57: Article 133
[27]	Simon L. Lectures on Geometric Measure Theory. Canberra: Australian National University Centre for Mathematical Analysis, 1983
[28]	Soner H M. Ginzburg-Landau equation and motion by mean curvature. I. Convergence J Geom Anal, 1997, 7: 437-475
[29]	Soner H M. Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface. J Geom Anal, 1997, 7: 477-491 doi: 10.1007/BF02921629
[30]	Takasao K, Tonegawa Y. Existence and regularity of mean curvature flow with transport term in higher dimensions. Math Ann, 2016, 364: 857-935 doi: 10.1007/s00208-015-1237-5
[31]	Tonegawa Y. Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math J, 2003, 33: 323-341