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 ...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.展开更多
基金Partially supported by the National Natural Science Foundation of Chinathe Grant of Chinese State Education Committee.
文摘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.
