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 à bord . 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.
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 with a smooth boundary . 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.
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@article{CRMATH_2017__355_2_161_0, author = {Meas, Len}, title = {Dispersive estimates for the wave equation inside cylindrical convex domains: {A} model case}, journal = {Comptes Rendus. Math\'ematique}, pages = {161--165}, publisher = {Elsevier}, volume = {355}, number = {2}, year = {2017}, doi = {10.1016/j.crma.2017.01.005}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2017.01.005/} }
TY - JOUR AU - Meas, Len TI - Dispersive estimates for the wave equation inside cylindrical convex domains: A model case JO - Comptes Rendus. Mathématique PY - 2017 SP - 161 EP - 165 VL - 355 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2017.01.005/ DO - 10.1016/j.crma.2017.01.005 LA - en ID - CRMATH_2017__355_2_161_0 ER -
%0 Journal Article %A Meas, Len %T Dispersive estimates for the wave equation inside cylindrical convex domains: A model case %J Comptes Rendus. Mathématique %D 2017 %P 161-165 %V 355 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2017.01.005/ %R 10.1016/j.crma.2017.01.005 %G en %F CRMATH_2017__355_2_161_0
Meas, Len. Dispersive estimates for the wave equation inside cylindrical convex domains: A model case. Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 161-165. doi : 10.1016/j.crma.2017.01.005. http://archive.numdam.org/articles/10.1016/j.crma.2017.01.005/
[1] On Strichartz estimates for Schrödinger operators in compact manifolds with boundary, Proc. Amer. Math. Soc., Volume 130 (2008), pp. 247-256
[2] Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 1817-1829
[3] The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Class. Math., Springer-Verlag, New York, 2003
[4] Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. Math. (2), Volume 180 (2014), pp. 323-380
[5] 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.