Double logarithmic inequality with a sharp constant

Ibrahim, S.; Majdoub, M.; Masmoudi, N.

doi:10.1090/S0002-9939-06-08240-2

Double logarithmic inequality with a sharp constant
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by S. Ibrahim, M. Majdoub and N. Masmoudi PDF

Proc. Amer. Math. Soc. 135 (2007), 87-97 Request permission

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

Shinji Adachi and Kazunaga Tanaka, Trudinger type inequalities in $\mathbf R^N$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. MR 1646323, DOI 10.1090/S0002-9939-99-05180-1
A. Alvino, G. Trombetti, and P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal. 13 (1989), no. 2, 185–220. MR 979040, DOI 10.1016/0362-546X(89)90043-6
Amel 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) 7 (2004), no. 1, 1–21 (English, with English and Italian summaries). MR 2044259
Philip Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z. 186 (1984), no. 3, 383–391. MR 744828, DOI 10.1007/BF01174891
Jean-Yves Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995), 177 (French, with French summary). MR 1340046
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), no. 4, 399–442 (English, with French summary). MR 824083
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.
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z. 231 (1999), no. 3, 479–487. MR 1704989, DOI 10.1007/PL00004737
M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 215–231. MR 1664497, DOI 10.3934/dcds.1999.5.215
Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\Bbb R^2$, J. Funct. Anal. 219 (2005), no. 2, 340–367. MR 2109256, DOI 10.1016/j.jfa.2004.06.013
Giorgio Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994) Prometheus, Prague, 1994, pp. 177–230. MR 1322313
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028