## Differential and Integral Equations

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

#### 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