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Binomial asymptotics of the spectrum of a boundary-value problem. (English. Russian original) Zbl 0583.35082
Funct. Anal. Appl. 17, 309-311 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 79-81 (1983).
Consider the elliptic boundary value problem of order 2m: \[ (*)\quad Au(x)=\lambda u(x)\quad on\quad \Omega \subseteq {\mathbb{R}}^ n,\quad Bj u(x)=0\quad on\quad \Gamma =\partial \Omega. \] Suppose (*) is self- adjoint, semi bounded below and let N(\(\lambda)\) be the distribution function of (*). In this paper the author extends previous results of V. Y. Ivrii: \[ N(\lambda)=\gamma \lambda^{n/2m}+O(\lambda^{(n- 1)/2m});\quad \lambda \to +\infty \] where \(\gamma\) is the usual constant. Furthermore under a condition on the Hamiltonian flow for the symbol of A he proves \[ N(\lambda)=\gamma \lambda^{n/2m}+o(\lambda^{(n- 1)/2m});\quad \lambda \to +\infty. \]
Reviewer: D.Robert

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
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