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 be an indexed family of connected open sets in , that shrinks to a tree as approaches zero. Let be the Neumann Laplacian and be the restriction of an function to . For , set . Under the assumption that all the edges of are line segments, and some additional conditions on , we show that the limit function satisfies a second-order ordinary differential equation on with Kirchhoff boundary conditions on each vertex of .

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