@article{AIHPC_2009__26_1_101_0,
author = {Helffer, Bernard and Hoffmann-Ostenhof, Thomas and Terracini, S.},
title = {Nodal Domains and Spectral Minimal Partitions},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
volume = {26},
year = {2009},
pages = {101-138},
doi = {10.1016/j.anihpc.2007.07.004},
mrnumber = {2483815},
zbl = {1171.35083},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPC_2009__26_1_101_0}
}
Helffer, B.; Hoffmann-Ostenhof, T.; Terracini, S. Nodal Domains and Spectral Minimal Partitions. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) pp. 101-138. doi : 10.1016/j.anihpc.2007.07.004. http://gdmltest.u-ga.fr/item/AIHPC_2009__26_1_101_0/
[1] , , Handbook of Mathematical Functions, Applied Math. Series, vol. 55, National Bureau of Standards, 1964.
[2] , Critical Points of Solutions of Elliptic Equations in Two Variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (2) (1988) 229-256. | Numdam | MR 939628 | Zbl 0649.35026
[3] , Nodal Lines of Eigenfunctions of the Fixed Membrane Problem in General Convex Domains, Comment. Math. Helv. 69 (1) (1994) 142-154. | MR 1259610 | Zbl 0838.35006
[4] , , , Nodal Domain Theorems À La Courant, Documenta Math. 9 (2004) 283-299. | MR 2117417 | Zbl 1067.35051
[5] , Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7, Pitman, 1980. | MR 572958 | Zbl 0436.35063
[6] , , N-Dimensional Shape Optimization Under Capacitary Constraints, J. Differential Equations 123 (1995) 504-522. | MR 1362884 | Zbl 0847.49029
[7] , , , Existence Results for Some Optimal Partition Problems, in: Monographs and Studies in Mathematics, vol. 8, 1998, pp. 571-579. | MR 1657219 | Zbl 0915.49006
[8] , , Shape Optimization for Dirichlet Problems: Relaxed Formulation and Optimality Conditions, Appl. Math. Optim. 23 (1991) 17-49. | MR 1076053 | Zbl 0762.49017
[9] , , An Existence Result for a Class of Shape Optimization Problems, Arch. Rat. Mech. Anal. 122 (1993) 183-195. | MR 1217590 | Zbl 0811.49028
[10] , , An Optimal Partition Problem for Eigenvalues, J. Sci. Comput. 31 (2007) 5-14. | MR 2304268 | Zbl 1123.65060
[11] , , , A Variational Problem for the Spatial Segregation of Reaction-Diffusion Systems, Indiana Univ. Math. J. 54 (3) (2005) 779-815. | MR 2151234 | Zbl 1132.35397
[12] , , , On a Class of Optimal Partition Problems Related to the Fucik Spectrum and to the Monotonicity Formula, Calc. Var. 22 (2005) 45-72. | MR 2105968 | Zbl 1132.35365
[13] , , Eigenvalue Variation II. Multidimensional Problems, J. Differential Equations 104 (2) (1993) 263-297. | MR 1231469 | Zbl 0807.34033
[14] , Graph Theory, Graduate Texts in Mathematics, vol. 173, second ed., Springer, 2000. | MR 1743598 | Zbl 0945.05002
[15] , , Elliptic Partial Differential Equations of Second Order, Springer, 1983. | MR 737190 | Zbl 0562.35001
[16] , , Algebraic Topology: a First Course, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1981, Advanced Book Program. | MR 643101 | Zbl 0498.55001
[17] , , On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations, Amer. J. Math. 75 (1953) 449-476. | MR 58082 | Zbl 0052.32201
[18] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains, Extended version, Preprint, Sept. 2005, mp_arc 05343. | MR 2324557
[19] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains. Preprint, April 2006, Moscow Math. J., in press. | MR 2324557 | Zbl 1125.35068
[20] B. Helffer, T. Hoffmann-Ostenhof, On nodal patterns and spectral optimal partitions, Unpublished Notes, December 2005.
[21] , , Local Properties of Solutions of Schrödinger Equations, Comm. Partial Differential Equations 17 (1992) 491-522. | MR 1163434 | Zbl 0783.35054
[22] , , , Interior Hölder Estimates for Solutions of Schrödinger Equations and the Regularity of Nodal Sets, Comm. Partial Differential Equations 20 (1995) 1241-1243. | MR 1335750 | Zbl 0846.35036
[23] , , , Bounds on the Multiplicity of Eigenvalues for Fixed Membranes, GAFA 9 (1999) 1169-1188. | MR 1736932 | Zbl 0949.35102
[24] , Perturbation Theory for Linear Operators, second ed., Springer, 1977. | MR 1335452 | Zbl 0342.47009
[25] , Remarks on Courant's Nodal Theorem, Comm. Pure Appl. Math. 9 (1956) 543-550. | MR 80861 | Zbl 0070.32604
[26] , Boundary Behavior of Conformal Maps, Grundlehren der Mathematischen Wissenshaften, vol. 295, Springer, 1992. | MR 1217706 | Zbl 0762.30001
[27] , , Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. | MR 493421 | Zbl 0401.47001
[28] , La Pemiére Valuer Propre Du Laplacien Pour Le Problème De Dirichlet, J. Math. Pures Appl. 62 (1983) 129-152. | MR 713393 | Zbl 0526.35060
[29] , Lower Semi-Continuity of Positive Quadratic Forms, Proc. Roy. Soc. Edinburgh 29 (1977) 267-273. | MR 512713 | Zbl 0442.47017
[30] , A Canonical Decomposition for Quadratic Forms With Applications to Monotone Convergence Theorems, J. Funct. Anal. 28 (1978) 377-385. | MR 500266 | Zbl 0413.47029
[31] , A Convergence Theorem for Dirichlet Forms With Applications to Boundary Value Problems With Varying Domains, Math. Z. 219 (1995) 275-287. | MR 1337221 | Zbl 0822.31005
[32] , On Optimal Shape Design, J. Math. Pures Appl. 72 (1993) 537-551. | MR 1249408 | Zbl 0849.49021
[33] , A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966. | JFM 48.0412.02 | MR 1349110 | Zbl 0174.36202
[34] , Continuity of the Eigenvalues of Self-Adjoint Operators With Respect to the Strong Operator Topology, Integral Equations Operator Theory 3 (1980) 138-142. | MR 570866 | Zbl 0476.47008
[35] , Monotone Continuity of the Spectral Resolution and the Eigenvalues, Proc. R. Soc. Edinburgh Sect. A 85 (1980) 131-136. | MR 566070 | Zbl 0448.47008
[36] , Stetige Abhängigkeit Der Eigenwerte Und Eigenfunktionen Elliptischer Differentialoperatoren Vom Gebiet, Math. Scand. 54 (1984) 51-69. | MR 753063 | Zbl 0526.35061