Given any elliptic system with t -independent coefficients in the upper-half space, we obtain representation and trace for conormal gradient of solutions natural classes boundary value problems Dirichlet Neumann types area integral control or non-tangential maximal control. The spaces are obtained a range which is parametrized by properties some Hardy spaces. This implies complete picture uniqueness vs solvability well-posedness.

Abstract We consider elliptic operators in divergence form with lower order terms of the $$Lu = -{{\textrm{div}}}(A \cdot \nabla u + b ) - c du$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mtext>div</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>·</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> <mml:mi>c</mml:mi> <mml:mi>d</mml:mi>...

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Ω</m:mi> <m:mo>⊊</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\Omega\subsetneq{\mathbb{R}}^{n+1}} be open and let μ some measure supported on <m:mo>∂</m:mo> <m:mo>⁡</m:mo> {\partial\Omega} such that <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mo>(</m:mo> <m:mi>B</m:mi> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo>)</m:mo> <m:mo>≤</m:mo> <m:mi>C</m:mi>...

Let Ω⊂Rn+1, n≥1, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω ε-approximable or satisfies suitable square-function Carleson measure estimate. particular, applies to the case when Ω=Rn+1∖E and E regular. Our results establish conjecture posed by Hofmann, Martell, Mayboroda, in which they proved converse statements. Here also obtain two additional criteria for uniform...

Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated matrix $A$ real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz satisfy suitable Carleson type estimates. In this paper we show if $L^*$ is transpose of $A$, then $\partialΩ$ $n$-rectifiable only every solution $Lu=0$ $L^*v=0$ $Ω$...

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, be a bounded open and connected set satisfying the corkscrew condition. Assume also that its boundary $\partial \Omega$ is uniformly $n$-rectifiable measure theoretic agrees with topological up to of $n$-dimensional Hausdorff zero. In this paper we study equivalence between solvability $(D_{p'})$, Dirichlet problem for Laplacian data in $L^{p'}(\partial \Omega)$, $(R_{p})$ (resp. $(\tilde R_{p})$), regularity Haj\l asz Sobolev space...

Abstract We show that, for disjoint domains in the euclidean space whose boundaries satisfy a nondegeneracy condition, mutual absolute continuity of their harmonic measures implies with respect to surface measure and rectifiability intersection boundaries. © 2017 Wiley Periodicals, Inc.

We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies with respect to surface measure and rectifiability intersection boundaries. This improves on our previous result which assumed that boundaries satisfied capacity density condition.

We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of compact set E ⊂ R d , ≥ 2, is greater than d+1 2 then distance ∆(E) = {|x -y| : x, y ∈ E} contains an interval.We prove this result for sets ∆ B (E) { x -y E}, where • metric induced by norm defined symmetric bounded convex body with smooth boundary everywhere non-vanishing Gaussian curvature.We also obtain some detailed estimates pertaining Radon-Nikodym derivative measure.

We characterize the boundedness of square functions in upper half-space with general measures. The short proof is based on an averaging identity over good Whitney regions.

The purpose of this article is to study extrapolation solvability for boundary value problems elliptic systems in divergence form on the upper half-space assuming De Giorgi-type conditions. We develop a method allowing treat each problem independently others. shall base our energy solutions, estimates layers, equivalence certain with interior control so that reduces one-sided Rellich inequality. Our then amounts extrapolating inequality using atomic Hardy spaces, interpolation and duality....

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface in nontangentially accessible domains Ahlfors regular boundaries. We prove this fails high dimensions if we relax the regularity assumption by showing that, for each |$d>1$|⁠, there exists a Reifenberg flat domain |$\Omega\subset \mathbb{R}^{d+1}$| |$\mathcal{H}^{d}(\partial\Omega)<\infty$| subset |$E\subset \partial\Omega$| positive yet zero |$\mathcal{H}^{d}$|-measure. In...

Abstract Let $\Omega \subset{{\mathbb{R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated matrix $A$ real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz satisfy suitable Carleson type estimates. In this paper we show if $L^*$ is transpose of $A$, then $\partial \Omega $ $n$-rectifiable only every solution $Lu=0$...

Abstract Given E ⊂ ℝ d , define the volume set of 𝒱( ) = {det( x 1 2 ,..., : j ∈ }. In 3 we prove that has positive Lebesgue measure if either Hausdorff dimension is greater than 13/5, or a product form B × with ℝ, dim ℋ ( &gt; 2/3, 1,2,3. We show same conclusion holds for Salem subsets - 1, and give applications to discrete combinatorial geometry.