Electron. J. Diff. Equ., Vol. 2011 (2011), No. 100, pp. 1-54.

Stability and approximations of eigenvalues and eigenfunctions of the Neumann Laplacian, part 3 Michael M. H. Pang

Abstract:
This article is a sequel to two earlier articles (one of them written jointly with R. Banuelos) on stability results for the Neumann eigenvalues and eigenfunctions of domains in with a snowflake type fractal boundary. In particular we want our results to be applicable to the Koch snowflake domain. In the two earlier papers we assumed that a domain which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of and those of its approximating subdomains satisfy a uniform bound for all sufficiently small t>0. The purpose of this paper is to extend the results in the two earlier papers to the situations where the approximating domains are not necessarily subdomains of . We then apply our results to the Koch snowflake domain when it is approximated from outside by a decreasing sequence of polygons.

Submitted November 5, 2010. Published August 7, 2011.
Math Subject Classifications: 35P05, 35P15.
Key Words: Stability; approximations; Neumann eigenvalues and eigenfunctions.

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 Michael M. H. Pang Department of Mathematics, University of Missouri-Columbia Columbia, MO 65211, USA email: pangm@missouri.edu