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March/April 2011 Maximum and antimaximum principles near the second eigenvalue
J. Fleckinger, J.-P. Gossez, F. de Thélin
Differential Integral Equations 24(3/4): 389-400 (March/April 2011). DOI: 10.57262/die/1356019038


We consider the Dirichlet problem $ (*)$ $-\Delta u = \mu u + f $ in $\Omega$, $u=0$ on $\partial \Omega$, with $\Omega$ either a bounded smooth convex domain in $\mathbb R^2$, or a ball or an annulus in $\mathbb R^N$. Let $\lambda_2$ be the second eigenvalue, with $\varphi_2$ an associated eigenfunction. Although the two nodal domains of $\varphi_2$ do not satisfy the interior ball condition, we are able to prove under suitable assumptions that, if $\mu$ is sufficiently close to $\lambda_2$, then the solution $u$ of $(*)$ also has two nodal domains which appear as small deformations of the nodal domains of $\varphi_2$. For $N=2$, use is made in the proof of several results relative to the Payne conjecture.


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J. Fleckinger. J.-P. Gossez. F. de Thélin . "Maximum and antimaximum principles near the second eigenvalue." Differential Integral Equations 24 (3/4) 389 - 400, March/April 2011.


Published: March/April 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35127
MathSciNet: MR2757466
Digital Object Identifier: 10.57262/die/1356019038

Primary: 35J25

Rights: Copyright © 2011 Khayyam Publishing, Inc.

Vol.24 • No. 3/4 • March/April 2011
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