Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient

In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: -D∆∅ - 2α∇m(x) · ∇∅ + V(x)∅ = λ ∅ in Ω , complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, ∂∅ ⁄ ∂n + β (x)∅ = 0 on ∂ Ω , where β ∈ C(∂ Ω) is allowed to be positive, sign-changing, or negative, and n(x) is the unit exterior normal to ∂ Ω at x. The domain Ω ⊂ ℝ^N is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C²(Ω), V ∈ C(Ω) are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞ . 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., β = 0 on ∂ Ω ) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue.