Nodal domains and spectral minimal partitions

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

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 1, page 101-138
  • ISSN: 0294-1449

How to cite


Helffer, B., Hoffmann-Ostenhof, T., and Terracini, S.. "Nodal domains and spectral minimal partitions." Annales de l'I.H.P. Analyse non linéaire 26.1 (2009): 101-138. <>.

author = {Helffer, B., Hoffmann-Ostenhof, T., Terracini, S.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {optimal partitions; eigenvalues; nodal domains; spectral minimal partitions},
language = {eng},
number = {1},
pages = {101-138},
publisher = {Elsevier},
title = {Nodal domains and spectral minimal partitions},
url = {},
volume = {26},
year = {2009},

AU - Helffer, B.
AU - Hoffmann-Ostenhof, T.
AU - Terracini, S.
TI - Nodal domains and spectral minimal partitions
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 1
SP - 101
EP - 138
LA - eng
KW - optimal partitions; eigenvalues; nodal domains; spectral minimal partitions
UR -
ER -


  1. [1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions, Applied Math. Series, vol. 55, National Bureau of Standards, 1964. Zbl0171.38503
  2. [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. Zbl0649.35026MR939628
  3. [3] Alessandrini G., Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv.69 (1) (1994) 142-154. Zbl0838.35006MR1259610
  4. [4] Ancona A., Helffer B., Hoffmann-Ostenhof T., Nodal domain theorems à la Courant, Documenta Math.9 (2004) 283-299. Zbl1067.35051MR2117417
  5. [5] Bandle C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7, Pitman, 1980. Zbl0436.35063MR572958
  6. [6] Bucur P., Zolesio J.P., N-dimensional shape optimization under capacitary constraints, J. Differential Equations123 (1995) 504-522. Zbl0847.49029MR1362884
  7. [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. Zbl0915.49006MR1657219
  8. [8] Buttazzo G., Dal Maso G., Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim.23 (1991) 17-49. Zbl0762.49017MR1076053
  9. [9] Buttazzo G., Dal Maso G., An existence result for a class of shape optimization problems, Arch. Rat. Mech. Anal.122 (1993) 183-195. Zbl0811.49028MR1217590
  10. [10] Cafferelli L.A., Fang Hua Lin, An optimal partition problem for eigenvalues, J. Sci. Comput.31 (2007) 5-14. Zbl1123.65060MR2304268
  11. [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. Zbl1132.35397MR2151234
  12. [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. Zbl1132.35365MR2105968
  13. [13] Dauge M., Helffer B., Eigenvalue variation II. Multidimensional problems, J. Differential Equations104 (2) (1993) 263-297. Zbl0807.34033MR1231469
  14. [14] Diestel R., Graph Theory, Graduate Texts in Mathematics, vol. 173, second ed., Springer, 2000. Zbl0945.05002MR1743598
  15. [15] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer, 1983. Zbl0562.35001MR737190
  16. [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. Zbl0498.55001MR643101
  17. [17] Hartman P., Wintner A., On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math.75 (1953) 449-476. Zbl0052.32201MR58082
  18. [18] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains, Extended version, Preprint, Sept. 2005, mp_arc 05343. Zbl1125.35068MR2324557
  19. [19] B. Helffer, T. Hoffmann-Ostenhof, Converse spectral problems for nodal domains. Preprint, April 2006, Moscow Math. J., in press. Zbl1125.35068MR2324557
  20. [20] B. Helffer, T. Hoffmann-Ostenhof, On nodal patterns and spectral optimal partitions, Unpublished Notes, December 2005. 
  21. [21] Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Local properties of solutions of Schrödinger equations, Comm. Partial Differential Equations17 (1992) 491-522. Zbl0783.35054MR1163434
  22. [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 Equations20 (1995) 1241-1243. Zbl0846.35036MR1335750
  23. [23] Hoffmann-Ostenhof T., Michor P., Nadirashvili N., Bounds on the multiplicity of eigenvalues for fixed membranes, GAFA9 (1999) 1169-1188. Zbl0949.35102MR1736932
  24. [24] Kato T., Perturbation Theory for Linear Operators, second ed., Springer, 1977. Zbl0342.47009MR1335452
  25. [25] Pleijel A., Remarks on Courant's nodal theorem, Comm. Pure Appl. Math.9 (1956) 543-550. Zbl0070.32604MR80861
  26. [26] Pommerenke C., Boundary Behavior of Conformal Maps, Grundlehren der Mathematischen Wissenshaften, vol. 295, Springer, 1992. Zbl0762.30001MR1217706
  27. [27] Reed M., Simon B., Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. Zbl0401.47001MR493421
  28. [28] Shimakura N., La pemiére valuer propre du Laplacien pour le problème de Dirichlet, J. Math. Pures Appl.62 (1983) 129-152. Zbl0526.35060MR713393
  29. [29] Simon B., Lower semi-continuity of positive quadratic forms, Proc. Roy. Soc. Edinburgh29 (1977) 267-273. Zbl0442.47017MR512713
  30. [30] Simon B., A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal.28 (1978) 377-385. Zbl0413.47029MR500266
  31. [31] Stollmann P., A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains, Math. Z.219 (1995) 275-287. Zbl0822.31005MR1337221
  32. [32] Sverak V., On optimal shape design, J. Math. Pures Appl.72 (1993) 537-551. Zbl0849.49021MR1249408
  33. [33] Watson G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966. Zbl0174.36202MR1349110JFM48.0412.02
  34. [34] Weidmann J., Continuity of the eigenvalues of self-adjoint operators with respect to the strong operator topology, Integral Equations Operator Theory3 (1980) 138-142. Zbl0476.47008MR570866
  35. [35] Weidmann J., Monotone continuity of the spectral resolution and the eigenvalues, Proc. R. Soc. Edinburgh Sect. A85 (1980) 131-136. Zbl0448.47008MR566070
  36. [36] Weidmann J., Stetige Abhängigkeit der Eigenwerte und Eigenfunktionen elliptischer Differentialoperatoren vom Gebiet, Math. Scand.54 (1984) 51-69. Zbl0526.35061MR753063

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