The Limiting Equation for Neumann Laplacians on Shrinking Domains

Date

2000-04-26

Authors

Saito, Yoshimi

Journal Title

Journal ISSN

Volume Title

Publisher

Southwest Texas State University, Department of Mathematics

Abstract

Let {Ω∊}0<∊ ≤1 be an indexed family of connected open sets in ℝ², that shrinks to a tree Γ as ∊ approaches zero. Let HΩ∊ be the Neumann Laplacian and ƒ∊ be the restriction of an L²(Ω₁) function to Ω∊. For z ∈ ℂ\ [0, ∞), set u∊ = (HΩ∊ - z)-1 ƒ∊. Under the assumption that all the edges of Γ are line segments, and some additional conditions on Ω∊, we show that the limit function u0 = lim∊→0u∊ satisfies a second-order ordinary differential equation on Γ with Kirchhoff boundary conditions on each vertex of Γ.

Description

Keywords

Neumann Laplacian, Tree, Shrinking domains

Citation

Saito, Y. (2000). The limiting equation for Neumann Laplacians on shrinking domains. Electronic Journal of Differential Equations, 2000(31), pp. 1-25.

Rights

Attribution 4.0 International

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