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Irregular but nonfractal drums andn-dimensional weyl conjecture
Authors:Hua Chen
Institution:(1) Department of Mathematics, Wuhan University, 430072 Wuhan, People's Republic of China
Abstract:In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set OHgr subR n (n ge 1) with irregular but nonfractal boundarypartOHgr. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN i (lambda)(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ i (lambda)=phiv(lambda) –N i (lambda) for largelambda, wherephiv(lambda) is the Weyl term. This implies that for the irregular but nonfractal drum OHgr, not only the volume |OHgr| n is spectral invariant but also the area of boundary |partOHgr| n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.
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