@article{AIHPC_2009__26_1_101_0, 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://www.numdam.org/articles/10.1016/j.anihpc.2007.07.004/} }
TY - JOUR 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 SP - 101 EP - 138 VL - 26 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2007.07.004/ DO - 10.1016/j.anihpc.2007.07.004 LA - en ID - AIHPC_2009__26_1_101_0 ER -
%0 Journal Article %A Helffer, B. %A Hoffmann-Ostenhof, T. %A Terracini, S. %T Nodal Domains and Spectral Minimal Partitions %J Annales de l'I.H.P. Analyse non linéaire %D 2009 %P 101-138 %V 26 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2007.07.004/ %R 10.1016/j.anihpc.2007.07.004 %G en %F 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) no. 1, pp. 101-138. doi : 10.1016/j.anihpc.2007.07.004. http://www.numdam.org/articles/10.1016/j.anihpc.2007.07.004/
[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 | Zbl
,[3] Nodal Lines of Eigenfunctions of the Fixed Membrane Problem in General Convex Domains, Comment. Math. Helv. 69 (1) (1994) 142-154. | MR | Zbl
,[4] Nodal Domain Theorems À La Courant, Documenta Math. 9 (2004) 283-299. | MR | Zbl
, , ,[5] Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7, Pitman, 1980. | MR | Zbl
,[6] N-Dimensional Shape Optimization Under Capacitary Constraints, J. Differential Equations 123 (1995) 504-522. | MR | Zbl
, ,[7] Existence Results for Some Optimal Partition Problems, in: Monographs and Studies in Mathematics, vol. 8, 1998, pp. 571-579. | MR | Zbl
, , ,[8] Shape Optimization for Dirichlet Problems: Relaxed Formulation and Optimality Conditions, Appl. Math. Optim. 23 (1991) 17-49. | MR | Zbl
, ,[9] An Existence Result for a Class of Shape Optimization Problems, Arch. Rat. Mech. Anal. 122 (1993) 183-195. | MR | Zbl
, ,[10] An Optimal Partition Problem for Eigenvalues, J. Sci. Comput. 31 (2007) 5-14. | MR | Zbl
, ,[11] A Variational Problem for the Spatial Segregation of Reaction-Diffusion Systems, Indiana Univ. Math. J. 54 (3) (2005) 779-815. | MR | Zbl
, , ,[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 | Zbl
, , ,[13] Eigenvalue Variation II. Multidimensional Problems, J. Differential Equations 104 (2) (1993) 263-297. | MR | Zbl
, ,[14] Graph Theory, Graduate Texts in Mathematics, vol. 173, second ed., Springer, 2000. | MR | Zbl
,[15] Elliptic Partial Differential Equations of Second Order, Springer, 1983. | MR | Zbl
, ,[16] 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] 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] Local Properties of Solutions of Schrödinger Equations, Comm. Partial Differential Equations 17 (1992) 491-522. | MR | Zbl
, ,[22] 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] Bounds on the Multiplicity of Eigenvalues for Fixed Membranes, GAFA 9 (1999) 1169-1188. | MR | Zbl
, , ,[24] Perturbation Theory for Linear Operators, second ed., Springer, 1977. | MR | Zbl
,[25] Remarks on Courant's Nodal Theorem, Comm. Pure Appl. Math. 9 (1956) 543-550. | MR | Zbl
,[26] Boundary Behavior of Conformal Maps, Grundlehren der Mathematischen Wissenshaften, vol. 295, Springer, 1992. | MR | Zbl
,[27] Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978. | MR | Zbl
, ,[28] La Pemiére Valuer Propre Du Laplacien Pour Le Problème De Dirichlet, J. Math. Pures Appl. 62 (1983) 129-152. | MR | Zbl
,[29] Lower Semi-Continuity of Positive Quadratic Forms, Proc. Roy. Soc. Edinburgh 29 (1977) 267-273. | MR | Zbl
,[30] A Canonical Decomposition for Quadratic Forms With Applications to Monotone Convergence Theorems, J. Funct. Anal. 28 (1978) 377-385. | MR | Zbl
,[31] A Convergence Theorem for Dirichlet Forms With Applications to Boundary Value Problems With Varying Domains, Math. Z. 219 (1995) 275-287. | MR | Zbl
,[32] On Optimal Shape Design, J. Math. Pures Appl. 72 (1993) 537-551. | MR | Zbl
,[33] A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1966. | JFM | MR | Zbl
,[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 | Zbl
,[35] Monotone Continuity of the Spectral Resolution and the Eigenvalues, Proc. R. Soc. Edinburgh Sect. A 85 (1980) 131-136. | MR | Zbl
,[36] Stetige Abhängigkeit Der Eigenwerte Und Eigenfunktionen Elliptischer Differentialoperatoren Vom Gebiet, Math. Scand. 54 (1984) 51-69. | MR | Zbl
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