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

2012/10/02 by Auscher, Pascal, Badr, Nadine, Haller-Dintelmann, Robert +1
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1210.0780

Abstract

We show that, under general conditions, the operator (-∇ ⋅ μ∇ +1)^1/2 with mixed boundary conditions provides a topological isomorphism between W^1,p_D(Ω) and L^p(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.

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