摘要
In this paper, we study the asymptotics of the spectrum of the Dirichlet （or Neumann） Laplacian in a bounded open set ΩR<sup>n</sup>（n≥1） with irregular but nonfractal boundary Ω. We give a partial resolution of the Weyl conjecture, i.e. for the counting function Ni（λ）（i=0: Dirichlet; i=i: Neumann）, we have got a precise estimate of the remainder term Ψ<sub>i</sub>（λ）=φ（λ）-N<sub>i</sub>（λ） for large λ, where φ（λ） is the Weyl term. This implies that for the irregular but nonfractal drum Ω, not only the volume |Ω|<sub>n</sub> is spectral invariant but also the area of boundary |Ω|<sub>n-1</sub> might be spectral invariant as well.
In this paper, we study the asymptotics of the spectrum of the Dirichlet （or Neumann） Laplacian in a bounded open set ΩR<sup>n</sup>（n≥1） with irregular but nonfractal boundary Ω. We give a partial resolution of the Weyl conjecture, i.e. for the counting function Ni（λ）（i=0: Dirichlet; i=i: Neumann）, we have got a precise estimate of the remainder term Ψ<sub>i</sub>（λ）=φ（λ）-N<sub>i</sub>（λ） for large λ, where φ（λ） is the Weyl term. This implies that for the irregular but nonfractal drum Ω, not only the volume |Ω|<sub>n</sub> is spectral invariant but also the area of boundary |Ω|<sub>n-1</sub> might be spectral invariant as well.
基金
Partially supported by the National Natural Science Foundation of China
the Grant of Chinese State Education Committee.
