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.
Bibliographical notePublisher Copyright:
© 2014 American Mathematical Society.
- AMuckenhoupt weights
- Dirichlet problem
- Divergence form elliptic equations
- Harmonic measure
- Layer potentials
- Non-tangential maximal function
- Square function