Comptes Rendus
Mathematical analysis/Partial differential equations
Dispersive estimates for the wave equation inside cylindrical convex domains: A model case
Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 161-165.

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain ΩR3 with a smooth boundary Ω. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [1,2]. Better estimates in strictly convex domains have been obtained in [4]. Our case of cylindrical domains is an extension of the result of [4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.

Dans ce travail, nous allons établir des estimations de dispersion locales en temps pour les solutions de l'équation des ondes dans un domaine cylindrique convexe ΩR3 à bord C Ω. Les estimations de dispersion sont classiquement utilisées pour prouver les estimations de Strichartz. Dans un domaine Ω général, des estimations de Strichartz non optimales ont été démontrées par Blair–Smith–Sogge [1,2]. De meilleures estimations ont été prouvées dans [4] lorsque Ω est strictement convexe. Le cas des domaines cylindriques que nous considérons ici généralise les resultats de [4] dans le cas où la courbure ≥0 dépend de l'angle d'incidence et s'annule dans certaines directions.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.01.005

Len Meas 1

1 Laboratoire Jean-Alexandre-Dieudonné, UMR CNRS 7351, Université de Nice, parc Valrose, 06108 Nice Cedex 02, France
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Len Meas. Dispersive estimates for the wave equation inside cylindrical convex domains: A model case. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 161-165. doi : 10.1016/j.crma.2017.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.005/

[1] M.D. Blair; H.F. Smith; C.D. Sogge On Strichartz estimates for Schrödinger operators in compact manifolds with boundary, Proc. Amer. Math. Soc., Volume 130 (2008), pp. 247-256

[2] M.D. Blair; H.F. Smith; C.D. Sogge Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 1817-1829

[3] L. Hörmander The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Class. Math., Springer-Verlag, New York, 2003

[4] O. Ivanovici; G. Lebeau; F. Planchon Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. Math. (2), Volume 180 (2014), pp. 323-380

[5] O. Ivanovici; R. Lascar; G. Lebeau; F. Planchon Dispersion for the wave equation inside strictly convex domains II: the general case, 2016 | arXiv

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This work was supported by the ERC project SCAPDE.

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