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

By P. Auscher, N. Badr, Robert Haller-Dintelmann, Joachim Rehberg. 2012

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We show that, under general conditions, the operator $${(-\nabla . \mu \nabla + 1)^{1/2}}$$(-∇.μ∇+1)1/2 with mixed boundary conditions provides a topological isomorphism between $${W^{1,p}_D(\Omega) {\rm and} L^p(\Omega)}$$WD1,p(Ω)andLp(Ω), for $$p \in ]1,2[ $$p∈]1,2[ if one presupposes that this isomorphism holds true for p = 2. The domain $${\Omega}$$Ω 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 $${\overline {\partial \Omega \setminus D}}$$∂Ω\D¯ the existence of bi-Lipschitzian boundary charts is required.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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