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

The limiting equation for Neumann Laplacians on shrinking domains

Yoshimi Saito

Abstract:
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.

Show me the PDF file (238K), TEX file, and other files for this article.

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

Return to the EJDE web page