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 |

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