Asymptotic Behavior of the Principal Eigenvalue of a Linear Second Order Elliptic Operator with Small/Large Diffusion Coefficient

Abstract

In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: $-D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ { in }\Omega, $ complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, $ \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ { on }\partial\Omega, $ where $\beta\in C(\partial\Omega)$ is allowed to be positive, sign-changing, or negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$. The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants $D>0$ and $\alpha>0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to0$ or $D\to\infty$. Our results, together with those of [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 61 (2012), pp. 45--80; A. Devinatz, R. Ellis, and A. Friedman, Indiana Univ. Math. J., 23 (1973/74), pp. 991--1011; and A. Friedman, Indiana U. Math. J., 22 (1973), pp. 1005--1015] where the Neumann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue.