Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 100, pp. 1-54.
Title: Stability and approximations of eigenvalues and eigenfunctions
of the Neumann Laplacian, part 3
Author: Michael M. H. Pang (Univ. of Missouri-Columbia, MO, USA)
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 $\mathbb{R}^2$ 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 $\Omega$. 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 07, 2011.
Math Subject Classifications: 35P05, 35P15.
Key Words: Stability; approximations; Neumann eigenvalues
and eigenfunctions.