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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 5(347), Pages 3–78 (Mi umn552)  

This article is cited in 42 scientific papers (total in 42 papers)

Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains

M. S. Agranovich

Moscow State Institute of Electronics and Mathematics

Abstract: Spectral boundary-value problems with discrete spectrum are considered for second-order strongly elliptic systems of partial differential equations in a domain $\Omega\subset\mathbb R^n$ whose boundary $\Gamma$ is compact and may be $C^\infty$, $C^{1,1}$, or Lipschitz. The principal part of the system is assumed to be Hermitian and to satisfy an additional condition ensuring that the Neumann problem is coercive. The spectral parameter occurs either in the system (then $\Omega$ is assumed to be bounded) or in a first-order boundary condition. Also considered are transmission problems in $\mathbb R^n\setminus\Gamma$ with spectral parameter in the transmission condition on $\Gamma$. The corresponding operators in $L_2(\Omega)$ or $L_2(\Gamma)$ are self-adjoint operators or weak perturbations of self-adjoint ones. Under some additional conditions a discussion is given of the smoothness, completeness, and basis properties of eigenfunctions or root functions in the Sobolev $L_2$-spaces $H^t(\Omega)$ or $H^t(\Gamma)$ of non-zero order $t$ as well as of localization and the asymptotic behaviour of the eigenvalues. The case of Coulomb singularities in the zero-order term of the system is also covered.

DOI: https://doi.org/10.4213/rm552

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English version:
Russian Mathematical Surveys, 2002, 57:5, 847–920

Bibliographic databases:

UDC: 517.98
MSC: Primary 35J25, 35J55; Secondary 35J20, 35J50, 35P99, 35J05
Received: 17.04.2002

Citation: M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains”, Uspekhi Mat. Nauk, 57:5(347) (2002), 3–78; Russian Math. Surveys, 57:5 (2002), 847–920

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