Differential and Integral Equations

Maximum and antimaximum principles near the second eigenvalue

J. Fleckinger, J.-P. Gossez, and F. de Thélin

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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.

Article information

Differential Integral Equations, Volume 24, Number 3/4 (2011), 389-400.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations


Fleckinger, J.; Gossez, J.-P.; de Thélin , F. Maximum and antimaximum principles near the second eigenvalue. Differential Integral Equations 24 (2011), no. 3/4, 389--400. https://projecteuclid.org/euclid.die/1356019038

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