The Lp boundary value problems on Lipschitz domains

Soit £ = -div (A(x)*) un operateur elliptique du second ordre a coefficients reels mesurables bornes symetriques sur R n ou sur un domaine a bord Lipschitzien, soumis a une condition au bord de type Dirichlet. Pour tout p > 2, nous obtenons une condition necessaire et suffisante pour que la transformee de *(£) - 1 / 2 soit bornee sur l'espace L P . A titre d'application, nous etablissons pour 1 < p < 3 + ∈, le caractere borne en norme L p des transformees de Riez d'operateurs a coefficients VMO sur les domaines a bord Lipschitzien. L'intervalle obtenu pour p est optimal. Nous etudions egalement si *(£) - 1 / 2 est borne dans les espaces L 2 a poids.

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Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

2 Smoothness spaces and Lipschitz domains 15 2.1 Graph Lipschitz domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Hardy spaces on graph Lipschitz surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Bounded Lipschitz domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Besov and Triebel-Lizorkin spaces in Lipschitz domains . . . . . . . . . . . . . . . . . . . . 29 2.5 Smoothness spaces on Lipschitz boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

/pdf/boundary-value-problems-for-the-stokes-system-in-arbitrary-3pshvar3hj.pdf

Boundary value problems for the Stokes system in arbitrary Lipschitz domains

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying \textit{a posteriori} a separate work on bounded domains.

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

We develop new solvability methods for divergence form second order, real and complex, elliptic systems  above Lipschitz graphs, with $L_2$ boundary data.    The coefficients $A$ may depend on all ...

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon<p<2(n-1)/(n-3)+\epsilon$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partial\Omega)$. In the case of the polyharmonic equation $\Delta^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2006/0013/0001/MRL-2006-0013-0001-a011.pdf

The L^p Dirichlet problem for elliptic systems on Lipschitz domains

Let $\Cal{L}=\text{div}A\nabla$ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation $\Cal{L}u=0$ in a bounded Lipschitz domain $\Omega$ of $\Bbb{R}^n$. We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ with boundary data in $L^p(\partial \Omega)$ and that of the $L^q$ regularity problem $(R)_q$ with boundary data in $W^{1,q}(\partial \Omega)$, where $1<p,q<\infty$. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2007/0014/0002/MRL-2007-0014-0002-a004.pdf

A relationship between the Dirichlet and Regularity Problems for elliptic equations

The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.

Zhongwei Shen

Papers

The L^p Dirichlet problem for elliptic systems on Lipschitz domains

A relationship between the Dirichlet and Regularity Problems for elliptic equations

The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

A relationship between the Dirichlet and Regularity Problems for elliptic equations