The square root problem for second order, divergence form operators with mixed boundary conditions on Lp

Abstract

We show that, under general conditions, the operator (−∇⋅μ∇+1)1/2 with mixed boundary conditions provides a topological isomorphism between W1,pD(Ω) and Lp(Ω), for p∈]1,2[ if one presupposes that this isomorphism holds true for p=2. The domain Ω 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 ∂Ω∖D¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ the existence of bi-Lipschitzian boundary charts is required.