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. <i>Electronic Journal of Differential Equations, 2000</i>(31), pp. 1-25.

Rights

Attribution 4.0 International

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