Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition

Abstract

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 H1,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.

The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition

“…This result was further refined in the paper [8] which considered the L p (∂Ω) Dirichlet problem under the assumption that (1.1) δ(X) −1 osc B δ(X)/2 (X) a ij 2 and δ(X) sup B δ(X)/2 (X) b i 2 are the densities of Carleson measures with small Carleson norms. A recent paper [10] has established similar results for the Neuman and Regularity boundary value problems.…”

Regularity theory for solutions to second order elliptic operators with complex coefficients and the L Dirichlet problem

“…We apply this regularity result to the question of solvability of L p Dirichlet problem for elliptic operators of this type. This part of the paper is motivated by the known results concerning boundary value problems for second order elliptic equations in divergence form, when the coefficients are real and satisfy a certain natural, minimal smoothness condition (refer [18,19,27]). The literature on solvability of boundary value problems for complex coefficient operators in R n is limited, except when the matrix A is of block form.…”

“…Instead, we assume the coefficients A and B satisfy a natural Carleson condition that has appeared in the literature so far only for real elliptic operators. ( [27], [18], and [19]). The Carleson condition on A, (1.12) below, holds uniformly on Lipschitz subdomains, and is thus a natural condition in the context of chord-arc domains as well.…”

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

“…(∇u ε ) * ∈ L 2 (∂Ω), (1.5) and the solution u ε satisfies the estimate 6) where C depends only on d, κ 1 , κ 2 , (σ, M) and the Lipschitz character of Ω.…”