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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 |
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. |
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 $\mathbb{R}^2$, 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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