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

M. S. Agranovich

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 5, pp. 847-920

Résumé

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.