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Asymptotics of the spectrum of problems with constraints. (English. Russian original) Zbl 0615.35063

Math. USSR, Sb. 57, 77-95 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 1, 73-89 (1986).
The author has considered the asymptotics of the spectrum of problems with two types of constraints: \[ (0.1)\quad A_{11}u_ 1+A_{12}u_ 2=tu_ 1,\quad A^*_{12}u_ 1-A_{22}u_ 2=0,\quad or\quad Au=tBu \] and \[ (0,2)\quad Au+F^*_ p=tu;\quad Fu=0 \] where A,F are matrix with dimension \(m\times m\), \(m_ 1\times m\), \(m_ 1\leq m\). He has obtained the following spectral distribution function \[ (0.3)\quad N(t)=c_ 0t^{n/2\ell}+O(t^{(n-\delta)/2\ell}) \] where \(\delta\in (0,1/3)\), \(\delta\in (0,1/2)\) or \(\delta\in (0,2/3).\)
For the first problem (0.1), he mainly proved that
Theorem 1.1. The spectrum of the equivalent problem consists of the isolated and the finite multiple positive eigenvalue.
Theorem 1.2. For arbitrary \(\delta\in (0,1/3)\), when \(t\to \infty\). \[ (1.1.3)\quad N(t)=c_ 0t^{n/2\ell_ 1}+O(t^{(n-\delta)/2\ell_ 1}) \] where \(c_ 0=n^{-1}(2\pi)^{-n}\int_{\Omega}dx \int_{S_{n- 1}}ds(\xi)sp(a_ 0)^{-n/\ell}(x,\xi).\)
Theorem 1.3. Under some conditions, the formula (1.1.3) is also true for all \(\delta\in (0,1/2)\) or \(\delta\in (0,2/3).\)
For the second problem (0.2), he has proved that
Theorem 2.1. Under some condition, a) if \(\Omega\) is a Lipschitz boundary domain, then for arbitrary \(\delta\in (0,1/2)\), \[ (2.9)\quad N(t)=c_ 0t^{n/2\ell}+O(t^{(n-\delta)/2\ell}). \] b) if \(\Omega\) is piecewise smooth domain, then the formula (2.9) is also true for all \(\delta\in (0,2/3)\).
Reviewer: Li Mingzhong

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
74F15 Electromagnetic effects in solid mechanics
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