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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 23 (6 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 13 (3 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 the reg ..."
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Cited by 9 (2 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 ∂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 6 (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 Boundary Problem in L^p and Hardy spaces for Laplace's Equation on a Lipschitz Domain
"... We study the boundary regularity of solutions of the mixed problem for Laplace's equation in a Lipschitz graph domain # whose boundary is decomposed as ## = 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 ..."
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Cited by 2 (1 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 # whose boundary is decomposed as ## = 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 (##), where N(#u) is the nontangential maximal function of the gradient of u. 1 Introduction In this paper, we consider the mixed boundary value problem for Laplace's equation in a domain # # R n , n # 3. We assume that the boundary of # is decomposed as ## = N # D, where N # D = #. The mixed problem is stated as follows. Given functions f N and f D , find a function u which satisfies # # # #u = 0 in #, u = f D on D, #u ## = f N on N , (1) where #u ## = #u # represents the outer normal derivative on ##. Problems of this kind, including ...
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 2 (0 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 ...
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 1 (0 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.
ON AREA INTEGRAL ESTIMATES FOR SOLUTIONS TO PARABOLIC SYSTEMS IN TIMEVARYING AND NONSMOOTH CYLINDERS
, 2007
"... In this paper we prove results relating the (parabolic) nontangential maximum operator and appropriate square functions in Lp for solutions to general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of timevarying ..."
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In this paper we prove results relating the (parabolic) nontangential maximum operator and appropriate square functions in Lp for solutions to general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of timevarying, nonsmooth infinite cylinders Ω. In particular we prove a global as well as a local and scale invariant equivalence between the parabolic nontangential maximal operator and appropriate square functions for solutions of our system. The novelty of our approach is that it is not based on singular integrals, the prevailing tool in the analysis of systems in nonsmooth domains. Instead the methods explored have recently proved useful in the analysis of elliptic measure associated to nonsymmetric operators through the work of KenigKochPipherToro and in the analysis of caloric measure without the use of layer potentials.