Nodal Domains and Spectral Minimal Partitions

Helffer, B.; Hoffmann-Ostenhof, T.; Terracini, S.

doi:10.1016/j.anihpc.2007.07.004

Helffer, B. ; Hoffmann-Ostenhof, T. ; Terracini, S.

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 1, pp. 101-138.

MR Zbl | 4 citations dans Numdam

DOI : 10.1016/j.anihpc.2007.07.004

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     author = {Helffer, B. and Hoffmann-Ostenhof, T. and Terracini, S.},
     title = {Nodal {Domains} and {Spectral} {Minimal} {Partitions}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {101--138},
     publisher = {Elsevier},
     volume = {26},
     number = {1},
     year = {2009},
     doi = {10.1016/j.anihpc.2007.07.004},
     mrnumber = {2483815},
     zbl = {1171.35083},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2007.07.004/}
}

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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) no. 1, pp. 101-138. doi : 10.1016/j.anihpc.2007.07.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2007.07.004/

Bibliographie
Cité par

[1] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, Applied Math. Series, vol. 55, National Bureau of Standards, 1964.

[2] Alessandrini G., 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 | Zbl

[3] Alessandrini G., Nodal Lines of Eigenfunctions of the Fixed Membrane Problem in General Convex Domains, Comment. Math. Helv. 69 (1) (1994) 142-154. | MR | Zbl

[4] Ancona A., Helffer B., Hoffmann-Ostenhof T., Nodal Domain Theorems À La Courant, Documenta Math. 9 (2004) 283-299. | MR | Zbl

[5] Bandle C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7, Pitman, 1980. | MR | Zbl

[6] Bucur P., Zolesio J. P., N-Dimensional Shape Optimization Under Capacitary Constraints, J. Differential Equations 123 (1995) 504-522. | MR | Zbl

[7] Bucur P., Buttazzo G., Henrot A., Existence Results for Some Optimal Partition Problems, in: Monographs and Studies in Mathematics, vol. 8, 1998, pp. 571-579. | MR | Zbl

[8] Buttazzo G., Dal Maso G., Shape Optimization for Dirichlet Problems: Relaxed Formulation and Optimality Conditions, Appl. Math. Optim. 23 (1991) 17-49. | MR | Zbl

[9] Buttazzo G., Dal Maso G., An Existence Result for a Class of Shape Optimization Problems, Arch. Rat. Mech. Anal. 122 (1993) 183-195. | MR | Zbl

[10] Cafferelli L. A., Fang Hua Lin , An Optimal Partition Problem for Eigenvalues, J. Sci. Comput. 31 (2007) 5-14. | MR | Zbl

[11] Conti M., Terracini S., Verzini G., A Variational Problem for the Spatial Segregation of Reaction-Diffusion Systems, Indiana Univ. Math. J. 54 (3) (2005) 779-815. | MR | Zbl

[12] Conti M., Terracini S., Verzini G., On a Class of Optimal Partition Problems Related to the Fucik Spectrum and to the Monotonicity Formula, Calc. Var. 22 (2005) 45-72. | MR | Zbl

[13] Dauge M., Helffer B., Eigenvalue Variation II. Multidimensional Problems, J. Differential Equations 104 (2) (1993) 263-297. | MR | Zbl

[14] Diestel R., Graph Theory, Graduate Texts in Mathematics, vol. 173, second ed., Springer, 2000. | MR | Zbl

[15] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer, 1983. | MR | Zbl

[16] Greenberg M. J., Harper J. R., Algebraic Topology: a First Course, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1981, Advanced Book Program. | MR | Zbl

[17] Hartman P., Wintner A., On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations, Amer. J. Math. 75 (1953) 449-476. | MR | Zbl

[18] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains, Extended version, Preprint, Sept. 2005, mp_arc 05343. | MR

[19] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains. Preprint, April 2006, Moscow Math. J., in press. | MR | Zbl

[20] B. Helffer, T. Hoffmann-Ostenhof, On nodal patterns and spectral optimal partitions, Unpublished Notes, December 2005.

[21] Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Local Properties of Solutions of Schrödinger Equations, Comm. Partial Differential Equations 17 (1992) 491-522. | MR | Zbl

[22] Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Nadirashvili N., Interior Hölder Estimates for Solutions of Schrödinger Equations and the Regularity of Nodal Sets, Comm. Partial Differential Equations 20 (1995) 1241-1243. | MR | Zbl

[23] Hoffmann-Ostenhof T., Michor P., Nadirashvili N., Bounds on the Multiplicity of Eigenvalues for Fixed Membranes, GAFA 9 (1999) 1169-1188. | MR | Zbl

[24] Kato T., Perturbation Theory for Linear Operators, second ed., Springer, 1977. | MR | Zbl

[25] Pleijel A., Remarks on Courant's Nodal Theorem, Comm. Pure Appl. Math. 9 (1956) 543-550. | MR | Zbl

[26] Pommerenke C., Boundary Behavior of Conformal Maps, Grundlehren der Mathematischen Wissenshaften, vol. 295, Springer, 1992. | MR | Zbl

[27] Reed M., Simon B., Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. | MR | Zbl

[28] Shimakura N., La Pemiére Valuer Propre Du Laplacien Pour Le Problème De Dirichlet, J. Math. Pures Appl. 62 (1983) 129-152. | MR | Zbl

[29] Simon B., Lower Semi-Continuity of Positive Quadratic Forms, Proc. Roy. Soc. Edinburgh 29 (1977) 267-273. | MR | Zbl

[30] Simon B., A Canonical Decomposition for Quadratic Forms With Applications to Monotone Convergence Theorems, J. Funct. Anal. 28 (1978) 377-385. | MR | Zbl

[31] Stollmann P., A Convergence Theorem for Dirichlet Forms With Applications to Boundary Value Problems With Varying Domains, Math. Z. 219 (1995) 275-287. | MR | Zbl

[32] Sverak V., On Optimal Shape Design, J. Math. Pures Appl. 72 (1993) 537-551. | MR | Zbl

[33] Watson G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966. | JFM | MR | Zbl

[34] Weidmann J., Continuity of the Eigenvalues of Self-Adjoint Operators With Respect to the Strong Operator Topology, Integral Equations Operator Theory 3 (1980) 138-142. | MR | Zbl

[35] Weidmann J., Monotone Continuity of the Spectral Resolution and the Eigenvalues, Proc. R. Soc. Edinburgh Sect. A 85 (1980) 131-136. | MR | Zbl

[36] Weidmann J., Stetige Abhängigkeit Der Eigenwerte Und Eigenfunktionen Elliptischer Differentialoperatoren Vom Gebiet, Math. Scand. 54 (1984) 51-69. | MR | Zbl

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