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Zhongwei Shen - One of the best experts on this subject based on the ideXlab platform.
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L 2 Estimates in Lipschitz Domains
Periodic Homogenization of Elliptic Systems, 2018Co-Authors: Zhongwei ShenAbstract:In this chapter we study L2 boundary value problems for \({\mathcal{L}}_{\varepsilon}({u}_{\varepsilon}) = 0\) in a bounded Lipschitz Domain Ω.
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Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz Domains
arXiv: Analysis of PDEs, 2018Co-Authors: Zhongwei ShenAbstract: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 solvable for some $1
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Resolvent Estimates in L p for the Stokes Operator in Lipschitz Domains
Archive for Rational Mechanics and Analysis, 2012Co-Authors: Zhongwei ShenAbstract:We establish the L p resolvent estimates for the Stokes operator in Lipschitz Domains in $${\mathbb{R}^d}$$ , $${d\geqq 3}$$ for $${|\frac{1}{p}-\frac{1}{2}| < \frac{1}{2d} +\varepsilon}$$ . The result implies that the Stokes operator in a three-dimensional Lipschitz Domain generates a bounded analytic semigroup in L p for (3/2) − e < p < 3 + e. This gives an affirmative answer to a conjecture of M. Taylor (Progr. Nonlinear Differential Equations Appl., vol. 42, pp. 320–334).
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A bilinear estimate for biharmonic functions in Lipschitz Domains
Mathematische Annalen, 2010Co-Authors: Joel Kilty, Zhongwei ShenAbstract:We show that a bilinear estimate for biharmonic functions in a Lipschitz Domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz Domain Ω in \({\mathbb{R}^{d}}\) and 1 < q < ∞, the solvability of the L q Dirichlet problem for Δ 2 u = 0 in Ω with boundary data in WA 1,q (∂Ω) is equivalent to that of the L p regularity problem for Δ 2 u = 0 in Ω with boundary data in WA 2,p (∂Ω), where \({\frac{1}{p} + \frac{1}{q}=1}\). This duality relation, together with known results on the Dirichlet problem, allows us to solve the L p regularity problem for d ≥ 4 and p in certain ranges.
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A Bilinear Estimate for Biharmonic Functions in Lipschitz Domains
arXiv: Analysis of PDEs, 2009Co-Authors: Joel Kilty, Zhongwei ShenAbstract:We show that a bilinear estimate for biharmonic functions in a Lipschitz Domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded Lipschitz Domain $\Omega$ in $\rn{d}$ and $1
Michael Taylor - One of the best experts on this subject based on the ideXlab platform.
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Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem
Journal of Functional Analysis, 2000Co-Authors: Marius Mitrea, Michael TaylorAbstract:Abstract We continue a program to develop layer potential techniques for PDE on Lipschitz Domains in Riemannian manifolds. Building on L p and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz Domain in a Riemannian manifold with a metric tensor smooth of class C 1+ γ , for some γ >0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz Domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig.
M. S. Agranovich - One of the best experts on this subject based on the ideXlab platform.
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Mixed problems in a Lipschitz Domain for strongly elliptic second-order systems
Functional Analysis and Its Applications, 2011Co-Authors: M. S. AgranovichAbstract:We consider mixed problems for strongly elliptic second-order systems in a bounded Domain with Lipschitz boundary in the space ℝ^ n . For such problems, equivalent equations on the boundary in the simplest L _2-spaces H ^ s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _ p ^ s of Bessel potentials and Besov spaces B _ p ^ s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.
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remarks on potential spaces and besov spaces in a Lipschitz Domain and on whitney arrays on its boundary
Russian Journal of Mathematical Physics, 2008Co-Authors: M. S. AgranovichAbstract:In this note, we propose to remove some small gaps in the theory of potential spaces Hps(Ω) and Besov spaces Bps(Ω), 1 < p < ∞, s ∈ ℝ, for a bounded Lipschitz Domain Ω ⊂ ℝn, n ⩾ 2. Namely, we discuss 1) the unified definitions of these spaces with s of any sign, the unified duality theorems and interpolation relations, 2) the possibility of constructing a function in these spaces with given array of traces of its derivatives on the boundary.
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To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz Domains
Functional Analysis and Its Applications, 2007Co-Authors: M. S. AgranovichAbstract:For strongly elliptic Systems with Douglis-Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz Domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest L _2-Sobolev spaces H ^ σ . The regularity results are obtained in the potential spaces H _ p ^ σ and Besov spaces B _ p ^ σ . In the case of second-order Systems, the author’s results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered with the use of Whitney arrays.
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On a mixed Poincaré-Steklov type spectral problem in a Lipschitz Domain
Russian Journal of Mathematical Physics, 2006Co-Authors: M. S. AgranovichAbstract:We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz Domain. The boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed. The aim is to weaken the conditions sufficient for justifying the classical asymptotic formula for the eigenvalues. We show that it suffices to assume the boundary to be C ^1 in a neighborhood of the support of the weight outside a closed subset of zero measure.
Marius Mitrea - One of the best experts on this subject based on the ideXlab platform.
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On the Dirichlet and Regularity Problems for the Bi-Laplacian in Lipschitz Domains
Integral Methods in Science and Engineering Volume 1, 2009Co-Authors: Irina Mitrea, Marius MitreaAbstract:Recall that a Lipschitz Domain is a Domain whose boundary is locally given by graphs of Lipschitz functions. The formulation of, respectively, the Dirichlet and regularity problems for the Laplacian in a Lipschitz Domain \(\Omega \subset \mathbb{R}^n \) is $$ \begin{array}{*{20}c} {(D_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L^p (\partial \Omega ),} \\ \end{array}} \right.} & {(R_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L_1^p (\partial \Omega ),} \\ \end{array}} \right.} \\ \end{array} $$ (24.1) .
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THE NONLINEAR HODGE-NAVIER-STOKES EQUATIONS IN Lipschitz DomainS
Differential and integral equations, 2009Co-Authors: Marius Mitrea, Sylvie MonniauxAbstract:We investigate the Navier-Stokes equations in a suitable functional setting, in a three-dimensional bounded Lipschitz Domain, equipped with "free boundary" conditions. In this context, we employ the Fujita-Kato method and prove the existence of a local mild solution. Our approach makes essential use of the properties of the Hodge-Laplacian in Lipschitz Domains.
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Layer potentials and boundary value problems for Laplacian in Lipschitz Domains with data in quasi-Banach Besov spaces
Annali di Matematica Pura ed Applicata, 2006Co-Authors: Svetlana Mayboroda, Marius MitreaAbstract:We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz Domain $${\Omega}$$ , with boundary data in the Besov space $${B_{s}^{p,p} (\partial\Omega).}$$ The novelty is to identify a way of measuring smoothness for the solution u that allows us to consider the case p < 1. This is accomplished by using a Besov-based nontangential maximal function in place of the classical one. This builds on the works of Jerison and Kenig [ 14 ], where the case p > 1 was treated.
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Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem
Journal of Functional Analysis, 2000Co-Authors: Marius Mitrea, Michael TaylorAbstract:Abstract We continue a program to develop layer potential techniques for PDE on Lipschitz Domains in Riemannian manifolds. Building on L p and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz Domain in a Riemannian manifold with a metric tensor smooth of class C 1+ γ , for some γ >0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz Domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig.
Emmanuel Russ - One of the best experts on this subject based on the ideXlab platform.
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Hardy spaces and divergence operators on strongly Lipschitz Domains of Rn
Journal of Functional Analysis, 2003Co-Authors: Pascal Auscher, Emmanuel RussAbstract:Abstract Let Ω be a strongly Lipschitz Domain of R n . Consider an elliptic second-order divergence operator L (including a boundary condition on ∂Ω ) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L1. Under suitable assumptions on L, we identify this maximal Hardy space with H 1 ( R n ) if Ω= R n , with H r 1 (Ω) under the Dirichlet boundary condition, and with H z 1 (Ω) under the Neumann boundary condition.