Minimizing Movements for Mean Curvature Flow of Partitions

We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

[1]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[2]  Minimizing movements for mean curvature flow of droplets with prescribed contact angle , 2016, Journal de Mathématiques Pures et Appliquées.

[3]  On a relative Alexandrov-Fenchel inequality for convex bodies in Euclidean spaces , 2006 .

[4]  M. Gurtin,et al.  Fundamental contributions to the continuum theory of evolving phase interfaces in solids : a collection of reprints of 14 seminal papers, dedicated to Morton E. Gurtin on the occasion of his sixty-fifth birthday , 1999 .

[5]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[6]  M. Novaga,et al.  Evolution of networks with multiple junctions , 2016, 1611.08254.

[7]  A. Freire.  MEAN CURVATURE MOTION OF GRAPHS WITH CONSTANT CONTACT ANGLE AT A FREE BOUNDARY , 2008, 0812.1573.

[8]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[9]  Felix Otto,et al.  Threshold Dynamics for Networks with Arbitrary Surface Tensions , 2015 .

[10]  A. Chambolle.  An algorithm for Mean Curvature Motion , 2004 .

[11]  E. Giorgi.  Movimenti di Partizioni , 1996 .

[12]  Kenneth A. Brakke,et al.  The motion of a surface by its mean curvature , 2015 .

[13]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[14]  T. Ilmanen.  Elliptic regularization and partial regularity for motion by mean curvature , 1994 .

[15]  Yoshihiro Tonegawa,et al.  On the existence of mean curvature flow with transport term , 2010 .

[16]  David George Caraballo.  A variational scheme for the evolution of polycrystals by curvature , 1997 .

[17]  Tim Laux,et al.  Convergence of thresholding schemes incorporating bulk effects , 2016, 1601.02467.

[18]  Y. Tonegawa,et al.  On the mean curvature flow of grain boundaries , 2015, 1511.02572.

[19]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[20]  H. Garcke,et al.  Mean Curvature Flow with Triple Junctions in Higher Space Dimensions , 2012, 1207.2351.

[21]  A. Kader,et al.  Evolution of grain boundaries in two-dimensional foam , 1999 .

[22]  S. Luckhaus,et al.  Implicit time discretization for the mean curvature flow equation , 1995 .

[23]  G. Huisken.  Flow by mean curvature of convex surfaces into spheres , 1984 .

[24]  F. Otto,et al.  Convergence of the thresholding scheme for multi-phase mean-curvature flow , 2016, 1602.05857.

[25]  Antonin Chambolle,et al.  Nonlocal Curvature Flows , 2014, Archive for Rational Mechanics and Analysis.

[26]  A. Freire.  Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions , 2008, 0809.0636.

[27]  William P. Minicozzi,et al.  A Course in Minimal Surfaces , 2011 .

[28]  G. Bellettini,et al.  Lecture Notes on Mean Curvature Flow: Barriers and Singular Perturbations , 2014 .

[29]  A. Chambolle,et al.  Crystalline Mean Curvature Flow of Convex Sets , 2006 .

[30]  Klaus Ecker,et al.  Regularity Theory for Mean Curvature Flow , 2003 .

[31]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .

[32]  Carlo Mantegazza,et al.  Lecture Notes on Mean Curvature Flow , 2011 .

[33]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[34]  F. Maggi.  Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory , 2012 .

[36]  I. Tamanini,et al.  Metric spaces of partitions and Caccioppoli partitions , 2002 .

[37]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[38]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[39]  F. Almgren,et al.  Curvature-driven flows: a variational approach , 1993 .