TY - JOUR

T1 - Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators

AU - Hofmann, Steve

AU - Kenig, Carlos

AU - Mayboroda, Svitlana

AU - Pipher, Jill

N1 - Publisher Copyright:
© 2014 American Mathematical Society.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - We consider divergence form elliptic operators (formula presented), defined in the half space (formula presented), where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu= 0, and we then combine these estimates with the method of “ϵ-approximability” to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A∞with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.

AB - We consider divergence form elliptic operators (formula presented), defined in the half space (formula presented), where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu= 0, and we then combine these estimates with the method of “ϵ-approximability” to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A∞with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.

KW - AMuckenhoupt weights

KW - Dirichlet problem

KW - Divergence form elliptic equations

KW - Harmonic measure

KW - Layer potentials

KW - Non-tangential maximal function

KW - Square function

KW - ∊-approximability

UR - http://www.scopus.com/inward/record.url?scp=84938495508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938495508&partnerID=8YFLogxK

U2 - 10.1090/s0894-0347-2014-00805-5

DO - 10.1090/s0894-0347-2014-00805-5

M3 - Article

AN - SCOPUS:84938495508

VL - 28

SP - 483

EP - 529

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 2

ER -