Spectral problems for second-order strongly elliptic systems in domains with smooth and nonsmooth boundaries.

*(English. Russian original)*Zbl 1057.35019
Russ. Math. Surv. 57, No. 5, 847-920 (2002); translation from Usp. Mat. Nauk 57, No. 5, 3-79 (2002).

Author’s 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 Lipshitz. 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 \(L^t(\Gamma)\) 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.

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 \(L^t(\Gamma)\) 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.

Reviewer: Georgi E. Karadzhov (Sofia)

##### MSC:

35P05 | General topics in linear spectral theory for PDEs |

35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35J20 | Variational methods for second-order elliptic equations |

35J50 | Variational methods for elliptic systems |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |