Differential and Integral Equations

Maximum and antimaximum principles near the second eigenvalue

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019038

Mathematical Reviews number (MathSciNet)
MR2757466

Zentralblatt MATH identifier
1240.35127

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

Citation

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


Export citation