Differential and Integral Equations

A formula for principal eigenvalues of Dirichlet periodic parabolic problems with indefinite weight

T. Godoy, U. Kaufmann, and S. Paczka

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

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain and let $m$ be a $T$-periodic function such that $m_{\mid \Omega \times \left( 0,T\right) }\in L^{r}\left( \Omega \times \left( 0,T\right) \right) $ for some $r>N+2$ and $\int_{0}^{T} \,{\rm esssup}\, _{x\in \Omega }m\left( x,t\right) dt>0.$ Let $\lambda _{1}\left( m\right) $ be the (unique) positive principal eigenvalue of the Dirichlet periodic parabolic problem $ Lu=\lambda mu$ in $\Omega \times \mathbb{R}$, $u=0$ on $\partial \Omega \times \mathbb{R},$ $u>0$ in $\Omega \times \mathbb{R}.$ We prove a formula for $\lambda _{1}\left( m\right) $ which is an analogous of the well known variational expression for principal eigenvalues of self-adjoint elliptic problems. As a direct consequence we obtain monotonicity results for $ \lambda _{1}\left( m\right) $ with respect to the domain $\Omega $ and with respect to the zero order coefficient of the differential operator $L$.

Article information

Source
Differential Integral Equations Volume 20, Number 12 (2007), 1405-1422.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039072

Mathematical Reviews number (MathSciNet)
MR2377024

Zentralblatt MATH identifier
1212.35192

Subjects
Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35B10: Periodic solutions 35P05: General topics in linear spectral theory

Citation

Godoy, T.; Kaufmann, U.; Paczka, S. A formula for principal eigenvalues of Dirichlet periodic parabolic problems with indefinite weight. Differential Integral Equations 20 (2007), no. 12, 1405--1422. https://projecteuclid.org/euclid.die/1356039072.


Export citation