Results 1  10
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20
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
Bounds of Riesz transforms on L p spaces for second order elliptic operators
 Ann. Inst. Fourier
"... Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Rie ..."
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Cited by 10 (1 self)
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Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.
The CalderónZygmund theory for elliptic problems with measure data
, 2007
"... Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density propertie ..."
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Cited by 8 (4 self)
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Abstract. We consider nonlinear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = µ, and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable CalderónZygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
CalderónZygmund estimates for higher order systems with p(x)growth
 Math. Z
, 2008
"... m u), D m ∫ 〈 ϕ 〉 dx = F  Ω p(x)−2 F, D m 〉 ϕ dx, for allϕ ∈ C ∞ c (Ω; RN), m> 1, with variable growth exponent p: Ω → (1, ∞) we prove that if F  p(·) ∈ L q loc (Ω) with 1 < q < n n−2 + δ, then Dmu  p(·) ∈ L q loc (Ω). We should note that we prove this implication both in the non–degenerate ..."
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Cited by 6 (3 self)
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m u), D m ∫ 〈 ϕ 〉 dx = F  Ω p(x)−2 F, D m 〉 ϕ dx, for allϕ ∈ C ∞ c (Ω; RN), m> 1, with variable growth exponent p: Ω → (1, ∞) we prove that if F  p(·) ∈ L q loc (Ω) with 1 < q < n n−2 + δ, then Dmu  p(·) ∈ L q loc (Ω). We should note that we prove this implication both in the non–degenerate (µ> 0) and in the degenerate case (µ = 0). 1.
THE L p DIRICHLET PROBLEM FOR ELLIPTIC SYSTEMS ON LIPSCHITZ DOMAINS
, 2004
"... Abstract. We develop a new approach to the L p Dirichlet problem via L 2 estimates and reverse Hölder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain Ω in Rn. For n ≥ 4 and 2 − ε < + ε, we establish the solvability of ..."
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Cited by 4 (2 self)
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Abstract. We develop a new approach to the L p Dirichlet problem via L 2 estimates and reverse Hölder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain Ω in Rn. For n ≥ 4 and 2 − ε < + ε, we establish the solvability of the Dirichlet problem with boundary data in p < 2(n−1) n−3 L p (∂Ω). In the case of the polyharmonic equation ∆ ℓ u = 0 with ℓ ≥ 2, the range of p is sharp if 4 ≤ n ≤ 2ℓ + 1. 1.
The L p boundary value problems on Lipschitz domains
 Adv. Math
"... Abstract. Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and 2(n−1) − ε < p < 2 where ε> 0 depends on Ω, n+1 we obtain the solvability of the ..."
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Cited by 3 (0 self)
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Abstract. Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and 2(n−1) − ε < p < 2 where ε> 0 depends on Ω, n+1 we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.
NECESSARY AND SUFFICIENT CONDITIONS FOR THE SOLVABILITY OF THE L p DIRICHLET PROBLEM ON LIPSCHITZ DOMAINS.
, 2005
"... Abstract. We study the homogeneous elliptic systems of order 2ℓ with real constant coefficients on Lipschitz domains in Rn, n ≥ 4. For any fixed p> 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data i ..."
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Cited by 3 (1 self)
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Abstract. We study the homogeneous elliptic systems of order 2ℓ with real constant coefficients on Lipschitz domains in Rn, n ≥ 4. For any fixed p> 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in Lp. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the Lp Dirichlet problem for n ≥ 4 and 2 −ε < p < 2(n−1) n−3 +ε. The range of p is known to be sharp if ℓ ≥ 2 and 4 ≤ n ≤ 2ℓ + 1. For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if ℓ = 2, and n = 2ℓ + 2 if ℓ ≥ 3. 1.
CRACK INITIATION IN ELASTIC BODIES
"... Abstract. In this paper we study the crack initiation in a hyperelastic body governed by a Griffith’s type energy. We prove that, during a load process through a time dependent boundary datum of the type t → tg(x) and in absence of strong singularities (this is the case of homogeneous isotropic mat ..."
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Cited by 2 (0 self)
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Abstract. In this paper we study the crack initiation in a hyperelastic body governed by a Griffith’s type energy. We prove that, during a load process through a time dependent boundary datum of the type t → tg(x) and in absence of strong singularities (this is the case of homogeneous isotropic materials) the crack initiation is brutal, i.e., a big crack appears after a positive time ti> 0. On the contrary, in presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts (largely expected by the experts of material science) for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional Z E(u, Γ): = f(x, ∇v) dx + kH 1 (Γ),
GRADIENT REGULARITY FOR ELLIPTIC EQUATIONS IN THE HEISENBERG GROUP
, 708
"... Abstract. We give dimensionfree regularity conditions for a class of possibly degenerate subelliptic equations in the Heisenberg group exhibiting superquadratic growth in the horizontal gradient; this solves an issue raised in [40], where only dimension dependent bounds for the growth exponent ar ..."
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Cited by 1 (0 self)
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Abstract. We give dimensionfree regularity conditions for a class of possibly degenerate subelliptic equations in the Heisenberg group exhibiting superquadratic growth in the horizontal gradient; this solves an issue raised in [40], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal pLaplacean operator, extending some regularity proven in [17]. In turn, the a priori estimates found are shown to imply the suitable local CalderónZygmund theory for the related class of nonhomogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the subelliptic setting a few classical nonlinear Euclidean results [30, 14], and to the nonlinear case estimates of the same nature that were available in the subelliptic setting only for solutions to linear equations.
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.