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 $L$.