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Let Ω be a Lipschitz domain in ℝn, 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 H1,p(∂Ω) and of the Neumann problem with Lp(∂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 Lp(∂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.
|Number of pages||50|
|Journal||Communications on Pure and Applied Mathematics|
|Early online date||13 Jun 2016|
|Publication status||Published - Jul 2017|
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- 2 Finished
Solvability of elliptic partial differential equations with rough coefficients; the boundary value problems
12/09/12 → 11/09/15
Centre for analysis and nonlinear differential equations
1/08/07 → 31/07/14
- School of Mathematics - Personal Chair of harmonic analysis and partial differential
Person: Academic: Research Active