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