The asymptotic behavior of the first eigenvalue of differential operators degenerating on the boundary

Abstract

When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form ε Σ a i j ∂ 2 / ∂ x i ∂ x i + Σ b i ∂ / ∂ x i \varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i} then results are known regarding the behavior of the principal eigenvalue λ = λ ε \lambda = {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 . These results are very sharp in case the vector ( b i ) ({b_i}) has a unique asymptotically stable point in the domain ω \omega where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of ω \omega is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of λ ε {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 are established.