Dispersive and Strichartz estimates for the wave equation in domains with boundary

Ivanovici, Oana¹ 

¹ Université de Nice Sophia-Antipolis, Laboratoire J.A.Dieudonné, Parc Valrose 06108 Nice Cedex 02 FRANCE

Journées équations aux dérivées partielles (2010), article no. 11, 19 p.

Résumé

In this note we consider a strictly convex domain $\Omega \subset {ℝ}^d$ of dimension $d\ge 2$ with smooth boundary $\partial \Omega \ne \varnothing$ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.

Détail

DOI : 10.5802/jedp.68

Bibliographie

[1] Blair, Matthew D.; Smith, Hart F.; Sogge, Christopher D. Strichartz estimates for the wave equation on manifolds with boundary, to appear in Ann.Inst.H.Poincaré, Anal.Non Liréaire | MR | mathdoc-id

[2] Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nicolay Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., Volume 126 (2004) no. 3, pp. 569-605 | Zbl | MR

[3] Burq, Nicolas; Lebeau, Gilles; Planchon, Fabrice Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc., Volume 21 (2008) no. 3, pp. 831-845 | MR

[4] Davies, E. B. The functional calculus, J. London Math. Soc. (2), Volume 52 (1995) no. 1, pp. 166-176 | Zbl | MR

[5] Eskin, Gregory Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math., Volume 32 (1977), pp. 17-62 | Zbl | MR

[6] Ginibre, J.; Velo, G. Generalized Strichartz inequalities for the wave equation, Partial differential operators and mathematical physics (Holzhau, 1994) (Oper. Theory Adv. Appl.), Volume 78, Birkhäuser, Basel, 1995, pp. 153-160 | MR

[7] Grieser, Daniel $L^p$ bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. Thesis, UCLA, 1992 (http://www.staff.uni-oldenburg.de/daniel.grieser/wwwpapers/diss.pdf)

[8] Ivanovici, Oana Counter example to Strichartz estimates for the wave equation in domains, 2008 (to appear in Math. Annalen, arXiv:math/0805.2901) | arXiv

[9] Ivanovici, Oana Counterexamples to the Strichartz estimates for the wave equation in domains II, 2009 http://www.citebase.org/... (arXiv:math/0903.0048) | MR

[10] Ivanovici, Oana; Planchon, Fabrice Square function and heat flow estimates on domains, 2008 (arXiv:math/0812.2733) | arXiv

[11] Kapitanskiĭ, L. V. Some generalizations of the Strichartz-Brenner inequality, Algebra i Analiz, Volume 1 (1989) no. 3, pp. 127-159 | Zbl | MR

[12] Keel, Markus; Tao, Terence Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980 | Zbl | MR

[13] Lebeau, Gilles Estimation de dispersion pour les ondes dans un convexe, Journées “Équations aux Dérivées Partielles” (Evian, 2006), 2006 (see http://www.numdam.org/numdam-bin/fitem?id=JEDP_2006____A7_0) | mathdoc-id

[14] Lindblad, Hans; Sogge, Christopher D. On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., Volume 130 (1995) no. 2, pp. 357-426 | Zbl | MR

[15] Nier, Francis A variational formulation of Schrödinger-Poisson systems in dimension $d\le 3$, Comm. Partial Differential Equations, Volume 18 (1993) no. 7-8, pp. 1125-1147 | Zbl | MR

[16] Oraevsky, A.N. Whispering-gallery waves, Quantum Electronics, Volume 32 (2002) no. 5, pp. 377-400

[17] Smith, Hart F. A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 3, pp. 797-835 | Zbl | MR | mathdoc-id

[18] Smith, Hart F.; Sogge, Christopher D. On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., Volume 8 (1995) no. 4, pp. 879-916 | Zbl | MR

[19] Smith, Hart F.; Sogge, Christopher D. On the $L^p$ norm of spectral clusters for compact manifolds with boundary, Acta Math., Volume 198 (2007) no. 1, pp. 107-153 | Zbl | MR

[20] Strichartz, Robert S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Volume 44 (1977) no. 3, pp. 705-714 | Zbl | MR

[21] Tataru, Daniel Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 419-442 (electronic) | Zbl | MR