Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Trace Hardy-Sobolev inequalities in cones

Author: A. I. Nazarov
Translated by: the author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 997-1006
MSC (2010): Primary 46E35
Published electronically: August 22, 2011
MathSciNet review: 2760091
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sharp constants are found for the trace Hardy-Sobolev inequalities in cones. The question as to whether these constants are attained is discussed.

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

  • [AH] W. Allegretto and Y. X. Huang, A Picone's identity for the $ p$-Laplacian and applications, Nonlinear Anal. 32 (1998), 819-830. MR 1618334 (99c:35051)
  • [Br] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417. MR 1100809 (92d:46088)
  • [E] J. F. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J. 37 (1988), 687-698. MR 0962929 (90a:46071)
  • [GR] I. S. Gradshtein and I. M. Ryzhik, Tables of integrals, series, and products, 5th ed., Nauka, Moscow, 1971; English transl., 4th ed., Acad. Press, New York-London, 1965. MR 0197789 (33:5952)
  • [Km] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I, Akad. Verlag, Leipzig, 1959. MR 0106302 (21:5036)
  • [Kw] B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Discrete Contin. Dynam. Systems 6 (2000), 683-690. MR 1757396 (2001c:35064)
  • [LL] E. H. Lieb and M. Loss, Analysis, Grad. Stud. in Math., vol. 14, Amer. Math. Soc., Providence, RI, 1997. MR 1415616 (98b:00004)
  • [Ls] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré. Anal. Non Linéaire 1 (1984), 109-145, 223-283. MR 0778970 (87e:49035a); MR 0778974 (87e:49035b)
  • [M] V. G. Maz'ya, Sobolev spaces, Leningrad. Univ., Leningrad, 1985; English transl., Springer-Verlag, Berlin, 1985. MR 0807364 (87g:46055); MR 0817985 (87g:46056)
  • [MC] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), no. 2, 309-323. MR 1369395 (97d:49045)
  • [Nt] B. Nazaret, Best constant in Sobolev trace inequalities on the half-space, Nonlinear Anal. 65 (2006), no. 10, 1977-1985. MR 2258478 (2007m:46047)
  • [N] A. I. Nazarov, Dirichlet and Neumann problems to critical Emden-Fowler type equations, J. Global Optim. 40 (2008), 289-303. MR 2373558 (2009c:35150)
  • [N1] -, Hardy-Sobolev inequalities in a cone, Probl. Mat. Anal., No. 31, Tamara Rozhkovskaya, Novosibirsk, 2005, pp. 39-46; English transl., J. Math. Sci. 132 (2006), no. 4, 419-427. MR 2197336 (2006k:26038)
  • [N2] -, The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers, Probl. Mat. Anal., No. 20, Tamara Rozhkovskaya, Novosibirsk, 2000, pp. 171-190; English transl., J. Math. Sci. 102 (2000), no. 5, 4473-4486. MR 1807067 (2001k:35065)
  • [P] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19-30. MR 0547524 (81c:58026)
  • [PS] G. Pólya and G. Szegő, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., No. 27, Princeton Univ. Press, Princeton, NJ, 1951. MR 0043486 (13:270d)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 46E35

Retrieve articles in all journals with MSC (2010): 46E35

Additional Information

A. I. Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia

Keywords: Trace Sobolev inequality, trace Hardy inequality, sharp constants
Received by editor(s): January 18, 2010
Published electronically: August 22, 2011
Additional Notes: Partially supported by RFBR (grant no. 08-01-00748) and by the grant NSh.4210.2010.1.
Dedicated: Dedicated to V. M. Babich on the occasion of his 80th birthday
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society