Uniform rectifiability and ε\varepsilon-approximability of harmonic functions in LpL^p

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hyt\"onen and A. Ros\'en and the first author, J. M. Martell and S. Mayboroda.Comment: 34 pages. v2: accepted version; introduction updated, details added and some proofs re-written. To appear in Annales de l'Institut Fourie

The Green function estimates for strongly elliptic systems of second order

We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local H\"{o}lder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.Comment: bibliography correcte