Given any elliptic system with t-independent coefficients in the upperhalf space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.
ABSTRACT. In the present paper we prove that for any open connected set Ω ⊂ R n+1 , n ≥ 1, and any E ⊂ ∂Ω with H n (E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case n = 1.
Let Ω ⊂ R n+1 , n ≥ 1, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is ε-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω = R n+1 \ E and E is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "S < N " estimates, and another in terms of a suitable corona decomposition involving harmonic measure.
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