Trudinger type inequalities in

and their best exponents

Authors:
Shinji Adachi and Kazunaga Tanaka

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2051-2057

MSC (1991):
Primary 46E35, 26D10

DOI:
https://doi.org/10.1090/S0002-9939-99-05180-1

Published electronically:
November 1, 1999

MathSciNet review:
1646323

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

**[1]**D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,*Ann. of Math.***128**(1988), 385-398. MR**89i:46034****[2]**R. A. Adams,*Sobolev Spaces*,*Academic Press, New York*1975. MR**56:9247****[3]**L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser,*Bull. Sc. Math.***110**(1986), 113-127. MR**88f:46070****[4]**D. E. Edmunds and A. A. Ilyin, Asymptotically sharp multiplicative inequalities,*Bull. London Math. Soc.***27**(1995), 71-74. MR**97b:26015****[5]**M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,*Comm. Math. Helvetici***67**(1992), 471-497. MR**93k:58073****[6]**H. Kozono, T. Ogawa and H. Sohr, Asymptotic behaviour in for weak solutions of the Navier-Stokes equations in exterior domains,*Manuscripta Math.***74**(1992), 253-275. MR**92k:35220****[7]**J. B. McLeod and L. A. Peletier, Observations on Moser's inequality,*Arch. Rat. Mech. Anal.***106**(1989), 261-285. MR**90d:26029****[8]**J. Moser, A sharp form of an inequality by N. Trudinger,*Indiana Univ. Math. J.***20**(1979), 1077-1092. MR**46:662****[9]**T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear

Schrödinger equations,*Nonlinear Anal.***14**(1990), 765-769. MR**91d:35203****[10]**T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,*J. Math. Anal. Appl.***155**(1991), 531-540. MR**92a:35147****[11]**T. Ozawa, On critical cases of Sobolev's inequalities,*J. Funct. Anal.***127**(1995) 259-269. MR**96c:46039****[12]**R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,*Indiana Univ. Math. J.***21**(1972), 841-842. MR**45:2466****[13]**M. Struwe, Critical points of embeddings of into Orlicz spaces,*Ann. Inst. Henri Poincaré, Analyse non linéaire***5**(1988), 425-464. MR**90c:35084****[14]**N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,*J. Math. Mech.***17**(1967) 473-484. MR**35:7121**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46E35,
26D10

Retrieve articles in all journals with MSC (1991): 46E35, 26D10

Additional Information

**Shinji Adachi**

Affiliation:
Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

Email:
kazunaga@mn.waseda.ac.jp

**Kazunaga Tanaka**

Affiliation:
Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

DOI:
https://doi.org/10.1090/S0002-9939-99-05180-1

Received by editor(s):
May 5, 1998

Received by editor(s) in revised form:
August 26, 1998

Published electronically:
November 1, 1999

Additional Notes:
The second author was partially supported by the Sumitomo Foundation (Grant No. 960354) and Waseda University Grant for Special Research Projects 97A-140, 98A-122.

Communicated by:
Christopher Sogge

Article copyright:
© Copyright 2000
American Mathematical Society