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

Abstract

<p>In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: -<i>D</i>∆∅ - 2α∇<i>m</i>(<i>x</i>) · ∇∅ + <i>V</i>(<i>x</i>)∅ = λ ∅ in Ω , complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, ∂∅ ⁄ ∂<i>n</i> + <i>β</i> (<i>x</i>)∅ = 0 on ∂ Ω , where <i>β</i> ∈ <i>C</i>(∂ Ω) is allowed to be positive, sign-changing, or negative, and <i>n</i>(<i>x</i>) is the unit exterior normal to ∂ Ω at <i>x</i>. The domain Ω ⊂ ℝ^<i>N</i> is bounded and smooth, the constants <i>D</i> > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and <i>m</i> ∈ <i>C</i>²(Ω), <i>V</i> ∈ <i>C</i>(Ω) are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient <i>D</i> → 0 or <i>D</i> → ∞ . Our results, together with those of [X. F. Chen and Y. Lou, <i>Indiana Univ. Math. J.</i>, 61 (2012), pp. 45-80; A. Devinatz, R. Ellis, and A. Friedman, <i>Indiana Univ. Math. J.</i>, 23 (1973/74), pp. 991-1011; and A. Friedman, <i>Indiana U. Math. J.</i>, 22 (1973), pp. 1005-1015] where the Neumann boundary case (i.e., <i>β</i> = 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.</p>