Related papers: Elliptic measures and Square function estimates on…

In this paper, we investigate the boundary behavior of solutions of divergence-form operators with an elliptic symmetric part and a $BMO$ anti-symmetric part. Our results will hold in non-tangentially accessible (NTA) domains; these general…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness…

In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $\Gamma$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher…

The present paper establishes that the Robin harmonic measure is quantitatively mutually absolutely continuous with respect to the surface measure on any Ahlfors regular set in any (quantifiably) connected domain for any elliptic operator.…

We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a…

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a…

For a linear elliptic operator with a singular drift that satisfies a finite Carleson measure condition, we prove that there exist `ample' sawtooth domains of the unit ball $B(0,1)\subset \R^{n+1}$ so that a BMO solvability assumption in…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is…

Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then…

We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients…

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$…

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in…

In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the…

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel…

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously,…

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In…

Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\Omega$ in $\mathbb{R}^d$ is…