Peng, Rui; Zhang, Guanghui; Zhou, Maolin

Author(s)

Peng, Rui

Zhang, Guanghui

Zhou, Maolin

Publication Date

2019

Abstract

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 ⊂ ℝN 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.

Citation

SIAM Journal on Mathematical Analysis, 51(6), p. 4724-4753

ISSN

1095-7154

0036-1410

Link

https://hdl.handle.net/1959.11/31760

Publisher

Society for Industrial and Applied Mathematics

Title

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

Type of document

Journal Article

Entity Type

Publication

Author(s)	Peng, Rui Zhang, Guanghui Zhou, Maolin
Publication Date	2019
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<sup><i>N</i></sup> 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><sup>2</sup>(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>
Citation	SIAM Journal on Mathematical Analysis, 51(6), p. 4724-4753
ISSN	1095-7154 0036-1410
Link	https://hdl.handle.net/1959.11/31760
Publisher	Society for Industrial and Applied Mathematics
Title	Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient
Type of document	Journal Article
Entity Type	Publication

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

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