# zbMATH — the first resource for mathematics

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

##### MSC:
 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:
##### References:
 [1] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag (1972). [2] B. M. Levitan, Mat. Sb.,35, No. 2, 267-316 (1954). [3] L. Hormander, Acta Math.,121, Nos. 3-4, 193-218 (1968). · Zbl 0164.13201 [4] J. J. Duistermaat and V. W. Guillemin, Invent. Math.,29, No. 1, 39-79 (1975). · Zbl 0307.35071 [5] R. Seeley, Am. J. Math.,102, No. 5, 869-902 (1980). · Zbl 0447.35029 [6] V. Ya. Ivrii, Dokl. Akad. Nauk SSSR,258, No. 5, 1045-1046 (1981). [7] V. Ya. Ivrii, Funkts. Anal. Prilozhen.,17, No. 1, 71-72 (1983). [8] J. Bruning, Math. Z.,137, No. 1, 75-85 (1974). · Zbl 0275.47030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.