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 $\varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i}$ then results are known regarding the behavior of the principal eigenvalue $\lambda = {\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$. These results are very sharp in case the vector $({b_i})$ has a unique asymptotically stable point in the domain $\omega$ where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of $\omega$ is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of ${\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$ are established.