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Nodal domains for an elliptic problem with the spectral parameter near the fourth eigenvalue

par Fleckinger, Jacqueline;Gossez, Jean-Pierre ;de Thélin, François
Référence Differential and integral equations, 25, 11-12, page (1189-1202)
Publication Publié, 2012-11

Article révisé par les pairs

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Résumé :

We consider the Dirichlet problem -Δu = μu + f in Ω u = 0 on ∂Ω , where Ω is a bounded smooth domain in ℝN. Let λ be an eigenvalue with Pdbl an associated eigenfunction. We study the following question (*): Assuming ∫ Ω fPdbl≠ 0, has u the same number of nodal domains as Pdbl; if μ is suffciently close to λ The answer to (*) is known to be affrmative in various cases; see [1], [5], and [6]. Here we study a specific situation where, on the contrary, the answer to (*) is not always affirmative: Ω = the unit disk in ℝ2 and λ=λ4=λ5.