Fractal drums and then-dimensional modified Weyl-Berry conjecture |
| |
Authors: | Chen Hua B D Sleeman |
| |
Institution: | (1) Department of Mathematics, Wuhan University, 430072 Wuhan, P.R. China;(2) Department of Mathematics and Computer Science, University of Dundee, DD1 4HN Dundee, U.K. |
| |
Abstract: | In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set ![OHgr](/content/u467257h75qu1868/xxlarge937.gif) R
n
(n 1) with fractal boundary ![part](/content/u467257h75qu1868/xxlarge8706.gif) of interior Minkowski dimension ![delta](/content/u467257h75qu1868/xxlarge948.gif) (n–1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the counting function N( ) (i.e. the number of positive eigenvalues less than ) as ![lambda](/content/u467257h75qu1868/xxlarge955.gif) + , which is of the form ![lambda](/content/u467257h75qu1868/xxlarge955.gif) /2 times a negative, bounded and left-continuous function of . This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn 2. In addition, we also obtain explicit upper and lower bounds on the second term ofN( ). |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|