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The Quarterly Journal of Mathematics 2000 51(3):313-342; doi:10.1093/qjmath/51.3.313
© 2000 by Oxford University Press
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Neumann laplacians on domains and operators on associated trees

W. D. Evans1,* and Yoshimi Sait2,§

1 School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, USA 2 Department of Mathematics, UAB, Birmingham, Alabama 35294, USA

Connections are established between the essential specta of the Neumann Laplacian on a domain {Omega} in R2 and a one-term operator of Sturm–Liouville type which is defined naturally on the skeleton, or a generalized ridge, {Gamma} of {Omega}, when {Gamma} is a tree. Horns, spirals, ‘rooms and passages’ and domains with fractal boundaries, like the Koch snowflake, are examples of such domains {Omega}. The analysis hinges on the existence of isometric maps between L2({Omega}), H1({Omega}) and weighted L2, H1 spaces defined on {Gamma} in terms of a Lipschitz map {tau} which projects {Omega} onto {Gamma}.

Received 28 October, 1998. Revised 31 May, 1999.

* E-mail: hmsri@uvvm.uvic.ca

§ E-mail: vu@math-1.sci.kuniv.edu.kw


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