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

GODOY, T.; KAUFMANN, U.; PACZKA, S.

DIFFERENTIAL AND INTEGRAL EQUATIONS

Año: 2007 vol. 20 p. 1405 - 1422

0893-4983

Let Ω⊂R^{N} be a smooth bounded domain and let m be a T-periodic function such that m_{∣Ω×(0,T)}∈L^{r}(Ω×(0,T)) for some r>N+2 and ∫₀^{T}esssup_{x∈Ω}m(x,t)dt>0. Let λ₁(m) be the (unique) positive principal eigenvalue of the Dirichlet periodic parabolic problem Lu=λmu in Ω×R, u=0 on ∂Ω×R, u>0 in Ω×R. We prove a formula for λ₁(m) which is an analogous of the well known variational expression for principal eigenvalues of selfadjoint elliptic problems. As a direct consequence we obtain monotonicity results for λ₁(m) with respect to the domain Ω and with respect to the zero order coefficient of the differential operator L.