Spectral minimal partitions of a sector

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

Spectral minimal partitions of a sector

Virginie Bonnaillie-Noël ^1, and Corentin Léna ^2,

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

Abstract
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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.

Keywords: nodal domains, numerical simulations, Spectral theory, finite element method., Aharonov-Bohm Hamiltonian, minimal partitions.

Mathematics Subject Classification: Primary: 35B05, 35J05, 49M25, 65F15, 65N25; Secondary: 65N3.

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 $\mathbb{R}^2$, 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: http://dlmf.nist.gov/.
[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: http://anum-maths.univ-rennes1.fr/melina.
[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.

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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 $\mathbb{R}^2$, 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: http://dlmf.nist.gov/.
[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: http://anum-maths.univ-rennes1.fr/melina.
[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.

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