Boundary value problems with rough boundary data

Motivated by problems with boundary noise or with dynamic conditions on the boundary, we study boundary value problems with data on the boundary which do not belong to the classical trace spaces.

To obtain unique solvability, one can consider a class of Sobolev spaces with anisotropic structure and obtain continuity of the boundary trace with values in Besov spaces of negative order. In this way, we obtain unique solvability for problems with rough boundary data in the half-space and, to some extent, in domains. The results can be applied to prove the generation of an analytic semigroup for problems with dynamic boundary conditions.