Dirichlet and Neumann exterior problems for the \(n\)-dimensional Laplace operator. An approach in weighted Sobolev spaces (Q1354978)

The authors present a solution to Dirichlet and Neumann boundary value problems for the Laplace equation on an exterior domain in \(\mathbb{R}^n\). Instead of imposing explicit asymptotic conditions at infinity they require that the solution be from an appropriate weighted Sobolev space with power-logarithmic weights. This appears advantageous for both the theoretical and numerical approach because it enables to control not only the behaviour of the gradient but also of the solution itself. The boundary of the domain is considered to be Lipschitz-continuous if \(p=2\) and of the class \(\mathcal C^{1,1}\) if \(p\neq2\).