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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2013

Injections de Sobolev probabilistes et applications

Résumé

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $\mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2( \mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(\mathbb{S}^d), p<+\infty$ spaces. We also prove similar results on tori $\mathbb{T}^d$. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure $0$ set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.
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Dates et versions

hal-00646749 , version 1 (30-11-2011)

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Citer

Nicolas Burq, Gilles Lebeau. Injections de Sobolev probabilistes et applications. Annales Scientifiques de l'École Normale Supérieure, 2013, 46 (6), pp.915-960. ⟨hal-00646749⟩
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