Maximum and antimaximum principles : beyond the first eigenvalue

Mardi 18 octobre 2016 14:00-15:00 - Jean-Pierre Gossez - Université Libre de Bruxelles

Résumé : Consider the Dirichlet problem

$$-\Delta u = \mu u + f ; in ; \Omega, \, u=0 ; on ; \partial \Omega,$$

with $\Omega$ a smooth bounded domain in $I ! !R^N$.
The well-known maximum and antimaximum principles give informations on the sign of the solution $u$ when the parameter $ \mu$ varies near the first eigenvalue $\lambda_1$ of the corresponding homogenous problem. Our purpose in this talk is to introduce an analogue of these two principles when $\mu$ varies near a higher eigenvalue $\lambda_k$. Nodal domains play a central role in our study, as well as, in some cases, the Payne conjecture relative to the nodal line of a second eigenfunction in the plane. (Joint work with J. Fleckinger and F. de Thélin from Toulouse, France).

Lieu : Salle 113-115 (Bâtiment 425)

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