Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 31, pp. 1-25.

Title: The limiting equation for Neumann Laplacians on shrinking domains 

Author: Yoshimi Saito (Univ. of Alabama at Birmingham, AL, USA)
         
Abstract: 
  Let $\{\Omega_{\epsilon} \}_{0 < \epsilon \le1}$ be an indexed 
  family of connected open sets in ${\mathbb R}^2$, that shrinks 
  to a tree $\Gamma$ as $\epsilon$ approaches zero.
  Let $H_{\Omega_{\epsilon}}$ be the Neumann Laplacian 
  and $f_{\epsilon}$ be the restriction of an $L^2(\Omega_1)$ function 
  to $\Omega_{\epsilon} $. 
  For $z \in {\mathbb C}\backslash [0, \infty)$, set 
  $u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon} $.
  Under the assumption that all the edges of $\Gamma$ are line segments,
  and some additional conditions on $\Omega_{\epsilon}$, 
  we show that the limit function $u_0 = \lim_{\epsilon\to 0} u_{\epsilon}$ 
  satisfies a second-order ordinary differential equation on $\Gamma$ with
  Kirchhoff boundary conditions on each vertex of $\Gamma $.

Submitted March 9, 2000. Published April 26, 2000.
Math Subject Classifications: 35J05, 35Q99.
Key Words: Neumann Laplacian; tree; shrinking domains.