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Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 61 (15 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
A new approach to absolute continuity of elliptic measure, with applications to nonsymmetric equations
 Adv. in Math
"... In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc ..."
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Cited by 41 (8 self)
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In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc
Estimates for the Stokes Operator in Lipschitz Domains
, 1995
"... : We study the Stokes operator A in a threedimensional Lipschitz domain \Omega\Gamma Our main result asserts that the domain of A is contained in W 1;p 0 (\Omega\Gamma "W 3=2;2(\Omega\Gamma for some p ? 3. Certain L 1 estimates are also established. Our results may be used to improve th ..."
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Cited by 25 (4 self)
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: We study the Stokes operator A in a threedimensional Lipschitz domain \Omega\Gamma Our main result asserts that the domain of A is contained in W 1;p 0 (\Omega\Gamma "W 3=2;2(\Omega\Gamma for some p ? 3. Certain L 1 estimates are also established. Our results may be used to improve the regularity of strong solutions of NavierStokes equations in nonsmooth domains. In the appendix we provide a simple proof of area integral estimates for solutions of Stokes equations. Introduction In a recent interesting paper, Deuring and von Wahl [DW] consider strong solutions of the nonstationary NavierStokes equations in\Omega \Theta (0; T ): 8 ! : @u @t = \Deltau \Gamma (u \Delta r)u \Gamma rß + f; div u = 0; with the initialDirichlet condition ( u(X; t) = 0 for (X; t) 2 @\Omega \Theta (0; T ); u(X; 0) = u 0 (X) X 2\Omega ; where\Omega is a bounded Lipschitz domain in R 3 . Based on the functional analytical approach of Fujita and Kato [FK] and the Rellich estimates of S...
THE REGULARITY AND NEUMANN PROBLEM FOR NONSYMMETRIC ELLIPTIC OPERATORS
, 2006
"... We will consider the Dirichlet problem Lu=0, in Ω ..."
The Mixed Boundary Problem in L^p and Hardy spaces for Laplace’s Equation on a Lipschitz Domain
 CONTEMPORARY MATHEMATICS
"... We study the boundary regularity of solutions of the mixed problem for Laplace’s equation in a Lipschitz graph domain Ω whose boundary is decomposed as ∂Ω = N ∪D, where N ∩D = ∅. For a subclass of these domains, we show that if the Neumann data g is in L p (N) and if the Dirichlet data f is in the ..."
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Cited by 8 (4 self)
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We study the boundary regularity of solutions of the mixed problem for Laplace’s equation in a Lipschitz graph domain Ω whose boundary is decomposed as ∂Ω = N ∪D, where N ∩D = ∅. For a subclass of these domains, we show that if the Neumann data g is in L p (N) and if the Dirichlet data f is in the Sobolev space L p,1 (D), for 1 < p < 2, then the mixed boundary problem has a unique solution u for which N(∇u) ∈ L p (∂Ω), where N(∇u) is the nontangential maximal function of the gradient of u.
The ∂Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions
 Duke Math. J
, 2001
"... On a bounded pseudoconvex domain � in C n with a plurisubharmonic Lipschitz defining function, we prove that the ¯∂Neumann operator is bounded on Sobolev (1/2)spaces. Let D be a bounded pseudoconvex domain in C n with the standard Hermitian metric. The ¯∂Neumann operator N for (p, q)forms is the ..."
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Cited by 8 (1 self)
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On a bounded pseudoconvex domain � in C n with a plurisubharmonic Lipschitz defining function, we prove that the ¯∂Neumann operator is bounded on Sobolev (1/2)spaces. Let D be a bounded pseudoconvex domain in C n with the standard Hermitian metric. The ¯∂Neumann operator N for (p, q)forms is the inverse of the complex Laplacian □ = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯∂, where ¯ ∂ is the maximal extension of the CauchyRiemann operator on (p, q)forms with L 2coefficients and ¯ ∂ ∗ is its Hilbert space adjoint. The
THE MIXED PROBLEM IN LIPSCHITZ DOMAINS WITH GENERAL DECOMPOSITIONS OF THE BOUNDARY
"... Abstract. This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ R n, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N, D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shap ..."
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Cited by 7 (3 self)
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Abstract. This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ R n, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N, D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0> 1 depending on the domain Ω such that for p in the interval (1, p0), the mixed problem with Neumann data in the space L p (N) and Dirichlet data in the Sobolev space W 1,p (D) has a unique solution with the nontangential maximal function of the gradient of the solution in L p (∂Ω). We also obtain results for p = 1 when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces. 1.
Spectral Problems for the Lamé System with Spectral Parameter in Boundary Conditions on Smooth or Nonsmooth Boundary
, 1999
"... . The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are inve ..."
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Cited by 5 (2 self)
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. The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are investigated: (1) S is C 1 ; (2) S is Lipschitz. INTRODUCTION In this paper we consider four spectral problems for the Lame system of linear elasticity, see (1.3). The system contains the frequency parameter !, which is a xed complex number with Re! > 0. The statements of Problems I{IV are given in Subsection 1.1. The spectral parameter enters into the boundary conditions (in Problems I, II) or transmission conditions (in Problems III, IV) on a closed connected surface S, which divides its complement into a bounded domain G + and an unbounded domain G . This surface is assumed to be innitely smooth in Section 1 and Lipschitz in Section 2. Our aim is to study the spectral properties ...
On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains
 Proc. Lond. Math. Soc
"... Abstract. We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W ⊂ R3. We study the case when W is an axially symmetric, convex, bounded domain satisfying the John conditi ..."
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Cited by 3 (2 self)
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Abstract. We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W ⊂ R3. We study the case when W is an axially symmetric, convex, bounded domain satisfying the John condition. The Cartesian coordinates (x, y, z) are chosen so that the mean free surface of the liquid lies in (x, z)plane and yaxis is directed upwards (yaxis is the axis of symmetry). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction ϕ there is a change of x, z coordinates by a rotation around yaxis so that ϕ is odd in xvariable. The second result of the paper gives the following monotonicity property of the fundamental eigenfunction ϕ. If ϕ is odd in xvariable then it is strictly monotonic in xvariable. This property has the following hydrodynamical meaning. If liquid oscillates freely with fundamental frequency according to ϕ then the free surface elevation of liquid is increasing along each line parallel to xaxis during one period of time and decreasing during the other half period. The proof of the second result is based on the method developed by D. Jerison and N. Nadirashvili for the hot spots problem for Neumann Laplacian. 1.