Dirichlet-to-Neumann maps on bounded Lipschitz domains

The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains

Consistent operator semigroups and their interpolation

Under a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators