Differential and Integral Equations

Maximum and antimaximum principles: beyond the first eigenvalue

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

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We consider the Dirichlet problem (*)$ -\bigtriangleup u=\mu u+f$ in $\Omega,u=0$ on $\partial\Omega$. Let $\widehat{\lambda}$ be an eigenvalue, with $\widehat{\varphi}$ an associated eigenfunction. Under suitable assumptions on $f$ and on the nodal domains of $\widehat{\varphi}$, we show that, if $\mu$ is sufficiently close to $\widehat{\lambda}$, then the solution $u$ of (*) has the same number of nodal domains as $\widehat{\varphi}$, and moreover the nodal domains of $u$ appear as small deformations of those of $\widehat{\varphi}$.

Article information

Differential Integral Equations, Volume 22, Number 9/10 (2009), 815-828.

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
Secondary: 35B50: Maximum principles 35P05: General topics in linear spectral theory


Fleckinger, J.; Gossez, J.-P.; de Thélin, F. Maximum and antimaximum principles: beyond the first eigenvalue. Differential Integral Equations 22 (2009), no. 9/10, 815--828. https://projecteuclid.org/euclid.die/1356019509

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