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Etude de sup u*inf u pour l'équation de la courbure scalaire prescrite en dimension >= 3 et inégalités de Harnack

Abstract : We study some apriori etimates of type sup*inf for solutions of scalar curvature equation on open set of R^n. We give some results when the exposant is subcritic and tend to critic Sobolev exposant, the perturbation of the critic exponent apear in our inequality and this result is true in all dimension >=3 with minimal condition on prescribed scalar curvatures. We also study the cases n=3 and 4, without subcritc perturbation. We obtain inequality of type (sup)^(1/3) * inf on any compact set of our open set. In dimension 4, we obtain uniform boundness of the sup if we suppose the min of solutions are uniformly bounded above. The cas when we have a parturbation of the equation by nonlinear term with subcritic terme is also studed and we obtain, in this cas the boudness on any compact set of our product sup * inf. In our work, we look the case of minoration of sup * int product on compact riemannian manifold of dimension >=2, for prescribed scalar curvature equation, we proove that if the scalar curvature is positive and not identiclly vanishing, the product sup * inf have a positive lower bound, in dimension >=3. In dimension 2, we have the same result for sup+inf if we suppose the gradient of prescribed scalar curvature is uniformly bounded. The cas of sphere, this condition is not necessely and the positivity and uniform boundness of prescribed scalar curvature is sufficent.
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https://tel.archives-ouvertes.fr/tel-00009722
Contributor : Samy Skander Bahoura <>
Submitted on : Saturday, July 9, 2005 - 6:58:20 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:40 PM
Long-term archiving on: : Tuesday, September 7, 2010 - 5:24:09 PM

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Samy Skander Bahoura. Etude de sup u*inf u pour l'équation de la courbure scalaire prescrite en dimension >= 3 et inégalités de Harnack. domain_other. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00009722⟩

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