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

Oana Ivanovici[1]

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

Journées Équations aux dérivées partielles (2010)

  • Volume: 347, Issue: 3, page 1-19
  • ISSN: 0752-0360

Abstract

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In this note we consider a strictly convex domain Ω ⊂ ℝ d of dimension d ≥ 2 with smooth boundary ∂ Ω ≠ ∅ 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.

How to cite

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Ivanovici, Oana. "Dispersive and Strichartz estimates for the wave equation in domains with boundary." Journées Équations aux dérivées partielles 347.3 (2010): 1-19. <http://eudml.org/doc/116377>.

@article{Ivanovici2010,
abstract = {In this note we consider a strictly convex domain $\Omega \subset \mathbb\{R\}^d$ of dimension $d\ge 2$ with smooth boundary $\partial \Omega \ne \emptyset$ 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.},
affiliation = {Université de Nice Sophia-Antipolis, Laboratoire J.A.Dieudonné, Parc Valrose 06108 Nice Cedex 02 FRANCE},
author = {Ivanovici, Oana},
journal = {Journées Équations aux dérivées partielles},
keywords = {Rayleigh whispering gallery modes; Dirichlet boundary condition},
language = {eng},
month = {6},
number = {3},
pages = {1-19},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Dispersive and Strichartz estimates for the wave equation in domains with boundary},
url = {http://eudml.org/doc/116377},
volume = {347},
year = {2010},
}

TY - JOUR
AU - Ivanovici, Oana
TI - Dispersive and Strichartz estimates for the wave equation in domains with boundary
JO - Journées Équations aux dérivées partielles
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
VL - 347
IS - 3
SP - 1
EP - 19
AB - In this note we consider a strictly convex domain $\Omega \subset \mathbb{R}^d$ of dimension $d\ge 2$ with smooth boundary $\partial \Omega \ne \emptyset$ 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.
LA - eng
KW - Rayleigh whispering gallery modes; Dirichlet boundary condition
UR - http://eudml.org/doc/116377
ER -

References

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  1. Matthew D. Blair, Hart F. Smith, Christopher D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, to appear in Ann.Inst.H.Poincaré, Anal.Non Liréaire Zbl1198.58012MR2566711
  2. Nicolas Burq, Patrick Gérard, Nicolay Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569-605 Zbl1067.58027MR2058384
  3. Nicolas Burq, Gilles Lebeau, Fabrice Planchon, Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc. 21 (2008), 831-845 Zbl1204.35119MR2393429
  4. E. B. Davies, The functional calculus, J. London Math. Soc. (2) 52 (1995), 166-176 Zbl0858.47012MR1345723
  5. Gregory Eskin, Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math. 32 (1977), 17-62 Zbl0375.35037MR477491
  6. J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, Partial differential operators and mathematical physics (Holzhau, 1994) 78 (1995), 153-160, Birkhäuser, Basel Zbl0839.35016MR1365328
  7. Daniel Grieser, L p bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. Thesis, UCLA, (1992) 
  8. Oana Ivanovici, Counter example to Strichartz estimates for the wave equation in domains, (2008) Zbl1201.35060
  9. Oana Ivanovici, Counterexamples to the Strichartz estimates for the wave equation in domains II, (2009) Zbl1201.35060MR2640046
  10. Oana Ivanovici, Fabrice Planchon, Square function and heat flow estimates on domains, (2008) Zbl1200.35066
  11. L. V. Kapitanskiĭ, Some generalizations of the Strichartz-Brenner inequality, Algebra i Analiz 1 (1989), 127-159 Zbl0732.35118MR1015129
  12. Markus Keel, Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980 Zbl0922.35028MR1646048
  13. Gilles Lebeau, Estimation de dispersion pour les ondes dans un convexe, Journées “Équations aux Dérivées Partielles” (Evian, 2006) (2006) 
  14. Hans Lindblad, Christopher D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357-426 Zbl0846.35085MR1335386
  15. Francis Nier, A variational formulation of Schrödinger-Poisson systems in dimension d ≤ 3 , Comm. Partial Differential Equations 18 (1993), 1125-1147 Zbl0785.35086MR1233187
  16. A.N. Oraevsky, Whispering-gallery waves, Quantum Electronics 32 (2002), 377-400 
  17. Hart F. Smith, A parametrix construction for wave equations with C 1 , 1 coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), 797-835 Zbl0974.35068MR1644105
  18. Hart F. Smith, Christopher D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995), 879-916 Zbl0860.35081MR1308407
  19. Hart F. Smith, Christopher D. Sogge, On the L p norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 (2007), 107-153 Zbl1189.58017MR2316270
  20. Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714 Zbl0372.35001MR512086
  21. Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), 419-442 (electronic) Zbl0990.35027MR1887639

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