文摘
We show that, under general conditions, the operator \({(-\nabla . \mu \nabla + 1)^{1/2}}\) with mixed boundary conditions provides a topological isomorphism between \({W^{1,p}_D(\Omega) {\rm and} L^p(\Omega)}\) , for \(p \in ]1,2[ \) if one presupposes that this isomorphism holds true for p?=?2. The domain \({\Omega}\) is assumed to be bounded, and the Dirichlet part D of the boundary has to satisfy the well-known Ahlfors–David condition, whilst for the points from \({\overline {\partial \Omega \setminus D}}\) the existence of bi-Lipschitzian boundary charts is required.