Electron. J. Diff. Eqns., Vol. 2000(2000), No. 31, pp. 1-25.

The limiting equation for Neumann Laplacians on shrinking domains

Yoshimi Saito

Let $\{\Omega_{\epsilon} \}_{0 < \epsilon \le1}$ be an indexed family of connected open sets in ${\Bbb R}^2$, 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 $z \in {\Bbb C}\backslash [0, \infty)$, 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.

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Yoshimi Saito
Department of mathematics
University of Alabama at Birmingham
Birmingham, AL 35294, USA.
e-mail: saito@math.uab.edu

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