Robin boundary value problems on arbitrary domains

We develop a theory of generalised solutions for elliptic boundary value pr oblems subject to Robin boundary conditions on arbitrary domains, which res embles in many ways that of the Dirichlet problem. In particular, we establ ish L-p-L-q-estimates which turn out to be the best possible in that framew ork. We also discuss consequences to the spectrum of Robin boundary value p roblems. Finally, we apply the theory to parabolic equations.