Related papers: Solvability of the Poisson-Dirichlet problem with …

On a bounded domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_{2,p}$ of functions $u$ whose Kenig-Pipher modified…

We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ \Omega = \mathcal O \times \mathbb R $, where the base $…

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection between the solvability of $(D_{p'})$,…

In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ where the coefficients of the matrix $A$ satisfy a…

Let $n\ge2$, $\Omega\subset\mathbb{R}^n$ be a bounded one-sided chord arc domain, and $p\in(1,\infty)$. In this article, we study the (weak) $L^p$ Poisson--Robin(-regularity) problem for a uniformly elliptic operator…

We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $\Omega$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $\Omega$ with…

We establish Dahlberg's perturbation theorem for non-divergence form operators L = A\nabla^2. If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p 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,…

In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a…

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…

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is…

We study the homogeneous elliptic systems of order $2\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse H\"older condition with exponent $p$ is necessary and sufficient…

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural algebraic condition, a strengthened version…

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in…

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the…

Let $\Omega\subset \mathbb R^{n+1}$, $n\geq2$, be an open set satisfying the corkscrew condition with $n$-Ahlfors regular boundary $\partial\Omega$, but without any connectivity assumption. We study the connection between solvability of the…