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

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 $\mathbb{R}^2$ 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 $\Omega\subseteq\mathbb{R}^2$ which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of $\Omega$ 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 $\Omega\subseteq\mathbb{R}^2$. 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

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