Overview

In this chapter boundary-value problems for the Laplace and Helmholtz operators are considered under weak assumptions on the smoothness of the domains. The theory we develop can be easily generalized to the case of uniformly elliptic formally self-adjoint differential operators with constant coefficients near infinity. We assume nothing about smoothness of the boundary S of a bounded domain D when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding i ₁ : H ¹( D) ? L ²( D) when the Neumann boundary condition is imposed; we assume boundedness of the embeddings i ₁ and of i ₂ : H ¹( D) ? L ²( S) when the Robin boundary condition is imposed, and, if, in addition, i ₁ and i ₂ are compact, then the boundary-value problems with the spectral parameter are of Fredholm type. Here i ₁ is the embedding of H ¹( D) (or H ¹( )) into L ²( D) ( L ²( )), D' := ? ⁿ \ D is the exterior domain, and is a bounded domain whose boundary consists of two components: S := ? D and , where is a smooth compact manifold. The space L ²( S) is the L ² space on S with respect to Hausdorff ( n - 1)-dimensional...