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arXiv: 1107.3826

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Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds

Nadine Badr 1, Frederic Bernicot 2, Emmanuel Russ 3, 4

(2011-07-19)

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat semigroup satisfying off-diagonal decays. We don't require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another one using functionals). We also give a chain rule and study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non bounded geometry) and even in the Euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

1:	Institut Camille Jordan (ICJ)
	CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon
2:	Laboratoire Paul Painlevé (LPP)
	CNRS : UMR8524 – Université Lille I - Sciences et technologies
3:	Laboratoire d'Analyse, Topologie, Probabilités (LATP)
	CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
4:	Institut Fourier (IF)
	CNRS : UMR5582 – Université Joseph Fourier - Grenoble I

Subject

:

Mathematics/Classical Analysis and ODEs

Keyword(s): Sobolev spaces – Riemannian manifold – algebra rule – paraproducts – heat semigroup

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From: Frederic Bernicot <>

Submitted on: Tuesday, 19 July 2011 18:02:11

Updated on: Tuesday, 19 July 2011 21:45:38