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Double logarithmic inequality with a sharp constant

Authors: S. Ibrahim, M. Majdoub and N. Masmoudi
Journal: Proc. Amer. Math. Soc. 135 (2007), 87-97
MSC (2000): Primary 49K20, 35L70
Published electronically: June 13, 2006
MathSciNet review: 2280178
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a Log Log inequality with a sharp constant. We also show that the constant in the Log estimate is ``almost'' sharp. These estimates are applied to prove a Moser-Trudinger type inequality for solutions of a 2D wave equation.

References [Enhancements On Off] (What's this?)

  • 1. S. Adachi and K. Tanaka: Trudinger type inequalities in $ \mathbb{R}^N$ and their best exponents, Proc. Amer. Math. Society, 128, N. 7, 2051-2057, 1999. MR 1646323 (2000m:46069)
  • 2. A. Alvino, P.-L. Lions and G. Trombetti: On optimization problem with prescribed rearrangements, Nonlinear Anal. T. M. A. 13, 185-220, 1989.MR 0979040 (90c:90236)
  • 3. A. Atallah Baraket: Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8, 1, 1-21, 2004. MR 2044259 (2005a:35198)
  • 4. P. Brenner: On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186, 383-391, 1984.MR 0744828 (85h:35183)
  • 5. J.-Y. Chemin: Fluides parfaits incompressibles, Astérisque no. 230, Société Mathématiques de France, 1995. MR 1340046 (97d:76007)
  • 6. L. C. Evans: Partial differential equations, Graduate Studies in Mathematics, AMS, 1998. MR 1625845 (99e:35001)
  • 7. J. Ginibre and G. Velo: Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrodinger equations, Ann. Inst. Poincaré Phys. Théo., 43, 399-442, 1985.MR 0824083 (87g:35208)
  • 8. S. Ibrahim, M. Majdoub and N. Masmoudi: Global solutions for a semilinear 2D Klein-Gordon equation with exponential type nonlinearity, to appear in Communications in Pure and Applied Mathematics.
  • 9. D. Kinderlehrer and G. Stampacchia: An introduction to variational inequalities and their applications, Academic Press, 1980. MR 0567696 (81g:49013)
  • 10. P.-L. Lions: The concentration-compactness principal in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1, 12-45, 1985.MR 0834360 (87c:49007)
  • 11. J. Moser: A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J., 20, 1077-1092, 1971. MR 0301504 (46:662)
  • 12. M. Nakamura and T. Ozawa: Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., 231, 479-487, 1999. MR 1704989 (2001b:35216)
  • 13. M. Nakamura and T. Ozawa: The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order, Discrete and Continuous Dynamical Systems, 5, no. 1, 215-231, 1999.MR 1664497 (99k:35128)
  • 14. B. Ruf: A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbb{R}^2$, J. Funct. Anal. 219 (2005), 340-367. MR 2109256 (2005k:46082)
  • 15. G. Talenti: Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, 5(Prague), 177-230, 1994.MR 1322313 (96a:46062)
  • 16. N.S. Trudinger: On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17, 473-484, 1967.MR 0216286 (35:7121)

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Additional Information

S. Ibrahim
Affiliation: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L8

M. Majdoub
Affiliation: Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire 1060, Tunis, Tunisia

N. Masmoudi
Affiliation: Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, New York 10012

Received by editor(s): January 9, 2005
Received by editor(s) in revised form: July 13, 2005
Published electronically: June 13, 2006
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2006 American Mathematical Society

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