## Differential and Integral Equations

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

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

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

#### 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