January  2014, 19(1): 27-53. doi: 10.3934/dcdsb.2014.19.27

Spectral minimal partitions of a sector

1. 

IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, F-35170 Bruz

2. 

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Bât. 425, F-91405 Orsay Cedex, France

Received  December 2012 Revised  July 2013 Published  December 2013

In this article, we are interested in determining spectral minimal $k$-partitions for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not nodal domains of eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of these minimal partitions. We first exhibit symmetric minimal partitions by using a mixed Dirichlet-Neumann Laplacian and then use a double covering approach to catch non symmetric candidates. In this way, we improve the known estimates of the energy associated with the minimal partitions.
Citation: Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27
References:
[1]

Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev., 115 1959, 485-491. doi: 10.1103/PhysRev.115.485.

[2]

B. Alziary, J. Fleckinger-Pellé and P. Takáč, Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in $\mathbbR^2$, Math. Methods Appl. Sci., 26 (2003), 1093-1136. doi: 10.1002/mma.402.

[3]

V. Bonnaillie-Noël and B. Helffer, Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions, Exp. Math., 20 (2011), 304-322. doi: 10.1080/10586458.2011.565240.

[4]

V. Bonnaillie-Noël, B. Helffer and T. Hoffmann-Ostenhof, Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions, J. Phys. A, 42 (2009), 185203, 20. doi: 10.1088/1751-8113/42/18/185203.

[5]

V. Bonnaillie-Noël, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions, ESAIM Control Optim. Calc. Var., 16 (2010), 221-246. doi: 10.1051/cocv:2008074.

[6]

D. Bucur, G. Buttazzo and A. Henrot, Existence results for some optimal partition problems, Adv. Math. Sci. Appl., 8 (1998), 571-579.

[7]

M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., 198 (2003), 160-196. doi: 10.1016/S0022-1236(02)00105-2.

[8]

M. Conti, S. Terracini and G. Verzini, On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations, 22 (2005), 45-72. doi: 10.1007/s00526-004-0266-9.

[9]

M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815. doi: 10.1512/iumj.2005.54.2506.

[10]

R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y. 1953.

[11]

E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.

[12]

NIST Digital Library of Mathematical Functions, Online companion to [20], Release 1.0.5 of 2012-10-01., Available from: , (). 

[13]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys., 202 (1999), 629-649. doi: 10.1007/s002200050599.

[14]

B. Helffer and T. Hoffmann-Ostenhof, On minimal partitions: New properties and applications to the disk, in Spectrum and Dynamics, 52 of CRM Proc. Lecture Notes, 119-135. Amer. Math. Soc., Providence, RI 2010.

[15]

B. Helffer and T. Hoffmann-Ostenhof, Minimal partitions for anisotropic tori, J. Spectr. Theory, (À paraître) (2013).

[16]

B. Helffer and T. Hoffmann-Ostenhof, On a magnetic characterization of spectral minimal partitions, Journal of the European Mathematical Society, (À paraître) (2013). doi: 10.4171/JEMS/415.

[17]

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 101-138. doi: 10.1016/j.anihpc.2007.07.004.

[18]

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, On spectral minimal partitions: the case of the sphere, in Around the Research of Vladimir Maz'ya. III, 13 of Int. Math. Ser. (N. Y.), 153-178. Springer, New York 2010. doi: 10.1007/978-1-4419-1345-6_6.

[19]

D. Martin, Mélina, Bibliothèque de Calculs éléments Finis,, 2007. Available from: , (). 

[20]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY 2010. Print Companion to [12].

[21]

K. Pankrashkin and S. Richard, Spectral and scattering theory for the Aharonov-Bohm operators, Rev. Math. Phys., 23 (2011), 53-81. doi: 10.1142/S0129055X11004205.

show all references

References:
[1]

Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev., 115 1959, 485-491. doi: 10.1103/PhysRev.115.485.

[2]

B. Alziary, J. Fleckinger-Pellé and P. Takáč, Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in $\mathbbR^2$, Math. Methods Appl. Sci., 26 (2003), 1093-1136. doi: 10.1002/mma.402.

[3]

V. Bonnaillie-Noël and B. Helffer, Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions, Exp. Math., 20 (2011), 304-322. doi: 10.1080/10586458.2011.565240.

[4]

V. Bonnaillie-Noël, B. Helffer and T. Hoffmann-Ostenhof, Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions, J. Phys. A, 42 (2009), 185203, 20. doi: 10.1088/1751-8113/42/18/185203.

[5]

V. Bonnaillie-Noël, B. Helffer and G. Vial, Numerical simulations for nodal domains and spectral minimal partitions, ESAIM Control Optim. Calc. Var., 16 (2010), 221-246. doi: 10.1051/cocv:2008074.

[6]

D. Bucur, G. Buttazzo and A. Henrot, Existence results for some optimal partition problems, Adv. Math. Sci. Appl., 8 (1998), 571-579.

[7]

M. Conti, S. Terracini and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., 198 (2003), 160-196. doi: 10.1016/S0022-1236(02)00105-2.

[8]

M. Conti, S. Terracini and G. Verzini, On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations, 22 (2005), 45-72. doi: 10.1007/s00526-004-0266-9.

[9]

M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815. doi: 10.1512/iumj.2005.54.2506.

[10]

R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y. 1953.

[11]

E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.

[12]

NIST Digital Library of Mathematical Functions, Online companion to [20], Release 1.0.5 of 2012-10-01., Available from: , (). 

[13]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys., 202 (1999), 629-649. doi: 10.1007/s002200050599.

[14]

B. Helffer and T. Hoffmann-Ostenhof, On minimal partitions: New properties and applications to the disk, in Spectrum and Dynamics, 52 of CRM Proc. Lecture Notes, 119-135. Amer. Math. Soc., Providence, RI 2010.

[15]

B. Helffer and T. Hoffmann-Ostenhof, Minimal partitions for anisotropic tori, J. Spectr. Theory, (À paraître) (2013).

[16]

B. Helffer and T. Hoffmann-Ostenhof, On a magnetic characterization of spectral minimal partitions, Journal of the European Mathematical Society, (À paraître) (2013). doi: 10.4171/JEMS/415.

[17]

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 101-138. doi: 10.1016/j.anihpc.2007.07.004.

[18]

B. Helffer, T. Hoffmann-Ostenhof and S. Terracini, On spectral minimal partitions: the case of the sphere, in Around the Research of Vladimir Maz'ya. III, 13 of Int. Math. Ser. (N. Y.), 153-178. Springer, New York 2010. doi: 10.1007/978-1-4419-1345-6_6.

[19]

D. Martin, Mélina, Bibliothèque de Calculs éléments Finis,, 2007. Available from: , (). 

[20]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY 2010. Print Companion to [12].

[21]

K. Pankrashkin and S. Richard, Spectral and scattering theory for the Aharonov-Bohm operators, Rev. Math. Phys., 23 (2011), 53-81. doi: 10.1142/S0129055X11004205.

[1]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617

[2]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120

[3]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[4]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561

[5]

Sarah Day; William D. Kalies; Konstantin Mischaikow and Thomas Wanner. Probabilistic and numerical validation of homology computations for nodal domains. Electronic Research Announcements, 2007, 13: 60-73.

[6]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[7]

Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks and Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113

[8]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic and Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253

[9]

Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems and Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93

[10]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

[11]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[12]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[13]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216

[14]

Panchi Li, Zetao Ma, Rui Du, Jingrun Chen. A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022002

[15]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[16]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[17]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[18]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[19]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[20]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]