American Institute of Mathematical Sciences

Advanced
July  2010, 28(3): 915-929. doi: 10.3934/dcds.2010.28.915

Analysis on the junctions of domain walls

Luis A. Caffarelli 1, and  Fang Hua Lin 2,

1. 

University of Texas at Austin, Department of Mathematics, 1 University Station, C1200, Austin, TX 78712-1082, United States
2. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, United States

Received  March 2010 Revised  April 2010 Published  April 2010

In this paper, we study the local structure and the smoothness of singularities of free boundaries in an optimal partition problem for the Dirichlet eigenvalues. We prove that there is a unique homogeneous blow up(tangent map) at each singular point in the interior of the free boundary. As a consequence we obtain the rectifiability as well as local structures of singularities.
Keywords: free boundaries; calculus of variations; montonicity formula..
Mathematics Subject Classification: 35R35; 49R9.
Citation: Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915
[1]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961
[2]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473
[3]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[4]

David Jerison, Nikola Kamburov. Free boundaries subject to topological constraints. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7213-7248. doi: 10.3934/dcds.2019301
[5]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[6]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[7]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[8]

Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2022, 14 (1) : 57-89. doi: 10.3934/jgm.2021012
[9]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[10]

Chonghu Guan, Xun Li, Zuo Quan Xu, Fahuai Yi. A stochastic control problem and related free boundaries in finance. Mathematical Control and Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021
[11]

Antonio Fasano, Mario Primicerio, Andrea Tesi. A mathematical model for spaghetti cooking with free boundaries. Networks and Heterogeneous Media, 2011, 6 (1) : 37-60. doi: 10.3934/nhm.2011.6.37
[12]

Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3651-3672. doi: 10.3934/cpaa.2020161
[13]

Yihong Du, Mingxin Wang, Meng Zhao. Two species nonlocal diffusion systems with free boundaries. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1127-1162. doi: 10.3934/dcds.2021149
[14]

Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313
[15]

Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098
[16]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
[17]

Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266
[18]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[19]

Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101
[20]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (119)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy