Sobolev Estimates for the Green Potential Associated with the Robin—Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition
We show that if u = Gλf is the solution operator for the Robin problem for the Laplacian, i.e., Δu = f in Ω, ∂ νu + λu = 0 on ∂Ω (with 0 ≤ λ ≤ ∞), then Gλ : Lp(Ω) → W 2,p(Ω) is bounded if 1 < p ≤ 2 and Ω ⊂ ℝn is a bounded Lipschitz domain satisfying a uniform exterior ball condition. This extends the earlier results of V. Adolfsson, B. Dahlberg, S. Fromm, D. Jerison, G. Verchota, and T. Wolff, who have dealt with Dirich-let (λ = ∞) and Neumann (λ = 0) boundary conditions. Our treatment of the end-point case p = 1 works for arbitrary Lipschitz domains and is conceptually different from the proof given by the aforementioned authors.
KeywordsBesov Space Neumann Problem Convex Domain Lipschitz Domain Robin Boundary Condition
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