Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition. Issue 7 (13th June 2016)

• Record Type: Journal Article
• Authors: Dindoš, Martin; Pipher, Jill; Rule, David
• Is Part Of: Communications on pure and applied mathematics. Volume 70:Issue 7(2017:Jul.)
• Page Start: 1316
• Page End: 1365
• Publication Date: 2016-06-13
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21649
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract: Let Ω be a Lipschitz domain in ℝ n, n ≥ 2, and L = div A ∇ be a second‐order elliptic operator in divergence form. We establish the solvability of the Dirichlet regularity problem with boundary data in H 1, p ( ∂ Ω ) and of the Neumann problem with L p ( ∂ Ω ) data for the operator L on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator L to be rough, obeying a certain Carleson condition with small norm. These results complete the results of Dindoš, Petermichl, and Pipher (2007), where the H 1, p ( ∂ Ω ) Dirichlet problem was considered under the same assumptions, and Dindoš and Rule (2010), where the regularity and Neumann problems were considered on two‐dimensional domains.© 2016 Wiley Periodicals, Inc.