Results 1  10
of
19
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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Paneitztype operators and applications
, 2000
"... Given (M,g) a smooth 4dimensional Riemannian manifold, let Sg be the scalar curvature of g, and let Rcg be the Ricci curvature of g. The Paneitz operator, discovered in [21], is the fourthorder operator defined by P 4 g u = �2 g u−divg ..."
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Cited by 28 (1 self)
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Given (M,g) a smooth 4dimensional Riemannian manifold, let Sg be the scalar curvature of g, and let Rcg be the Ricci curvature of g. The Paneitz operator, discovered in [21], is the fourthorder operator defined by P 4 g u = �2 g u−divg
On biharmonic maps and their generalizations
, 2003
"... We give a new proof of regularity of biharmonic maps from fourdimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions a ..."
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Cited by 13 (1 self)
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We give a new proof of regularity of biharmonic maps from fourdimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.
The Paneitz Curvature Problem on Lower Dimensional Spheres
, 2003
"... In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the nspheres, with n ∈ {5, 6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. ..."
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Cited by 12 (8 self)
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In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the nspheres, with n ∈ {5, 6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results.
Existence of conformal metrics on S n with prescribed fourthorder invariant
 Adv. Differential Equations
"... Abstract. In this paper we prescribe a fourth order conformal invariant on the standard nsphere, with n≥5, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbati ..."
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Cited by 9 (0 self)
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Abstract. In this paper we prescribe a fourth order conformal invariant on the standard nsphere, with n≥5, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the nonperturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry. 0. Introduction. Let (M 4, g) be a smooth 4dimensional manifold, Sg the scalar curvature of g, Ricg the Ricci curvature of g and ∆g = divg ∇g the LaplaceBeltrami operator on M 4. Let us consider the fourth order operator discovered by Paneitz [24] in 1983:
Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature
, 2003
"... In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the nspheres, with n ≥ 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given function defined on the sphere, ..."
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Cited by 5 (5 self)
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In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the nspheres, with n ≥ 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given function defined on the sphere, under which we prove existence of conformal metric with precribed Paneitz curvature.
On the conformal GaussBonnetChern inequality for LCF manifolds and related topics
 Calc. Var
"... Abstract. In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on evendimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q ..."
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Cited by 4 (0 self)
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Abstract. In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on evendimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Qcurvature is naturally related to the integrand in the classical GaussBonnetChern formula, i.e., the Pfaffian curvature. For a class of evendimensional complete LCF manifolds with integrable Qcurvature, we establish a GaussBonnetChern inequality. Second, a finiteness theorem for certain classes of complete LCF fourfold with integrable Pfaffian curvature is also proven. This is an extension of the classical results of CohnVossen and Huber in dimension two. It also can be viewed as a fully nonlinear analogue of results of ChangQingYang in dimension four. 1.
The sharp estimates for the first eigenvalue of Paneitz operator in 4manifold
, 2009
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Critical regularity for elliptic equations from LittlewoodPaley theory
, 2005
"... abstract. Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation ∆ α u + V u = 0 improves the Sobolev regularity of solutions provided the potential V is integrable with the critical power n/2α> 1. ..."
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Cited by 1 (0 self)
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abstract. Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation ∆ α u + V u = 0 improves the Sobolev regularity of solutions provided the potential V is integrable with the critical power n/2α> 1.