The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

Abstract

We show that, under general conditions, the operator $(-\nabla \cdot \mu \nabla +1{)}^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W_D^{1,p}(\Omega )$ 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, 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.

This version incorporates changes suggested by the referees