Year: 2004
Type: preprint
Abstract: In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding $i_{1}:H^{1}(D)\to L^{2}(D)$ when the Neumann boundary condition is imposed; we assume boundedness of the embeddings $i_{1}$ and of $i_{2}:H^{1}(D)\to L^{2}(\partial D)$ 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. Several examples of the classes of rough domains for which the embedding $i_2$ is compact are given. Applications to scattering by rough obstacles are mentioned.
Source: arXiv (Cornell University)
Authors: Goldshtein, V. G., Ramm, A. G.
Institutions: Ben-Gurion University of the Negev, Kansas State University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2127213447