Yu. Netrusov and Yu. Safarov
Weil asymptotic formula for the Laplacian on domains with rough boundaries
(105K, LaTeX)

ABSTRACT.  We study asymptotic distribution of eigenvalues of the Laplacian 
on a bounded domain in $R^n$. Our main results include an 
explicit remainder estimate in the Weyl formula for the Dirichlet 
Laplacian on an arbitrary bounded domain, sufficient conditions 
for the validity of the Weyl formula for the Neumann Laplacian on 
a domain with continuous boundary in terms of smoothness of the 
boundary and a remainder estimate in this formula. In particular, 
we show that the Weyl formula holds true for the Neumann Laplacian 
on a $\,\Lip_\alpha$-domain whenever $(d-1)/\alpha<d$, prove 
that the remainder in this formula is 
$O(\lambda^{(d-1)/\alpha})$ and give an example where the 
remainder estimate $O(\lambda^{(d-1)/\alpha})$ is order sharp. 
We use a new version of variational technique which does not 
require the extension theorem.