Trace Hardy–Sobolev inequalities in cones
HTML articles powered by AMS MathViewer
- by
A. I. Nazarov
Translated by: the author - St. Petersburg Math. J. 22 (2011), 997-1006
- DOI: https://doi.org/10.1090/S1061-0022-2011-01180-X
- Published electronically: August 22, 2011
- PDF | Request permission
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
- Walter Allegretto and Yin Xi Huang, A Picone’s identity for the $p$-Laplacian and applications, Nonlinear Anal. 32 (1998), no. 7, 819–830. MR 1618334, DOI 10.1016/S0362-546X(97)00530-0
- Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. MR 1100809, DOI 10.1002/cpa.3160440402
- José F. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J. 37 (1988), no. 3, 687–698. MR 962929, DOI 10.1512/iumj.1988.37.37033
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
- E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. 6. Aufl.; Teil II: Partielle Differentialgleichungen erster Ordnung für eine gesuchte Funktion. 4. Aufl, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 18, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1959 (German). MR 0106302
- Bernhard Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Discrete Contin. Dynam. Systems 6 (2000), no. 3, 683–690. MR 1757396, DOI 10.3934/dcds.2000.6.683
- Elliott H. Lieb and Michael Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. MR 1415616, DOI 10.1090/gsm/014
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145 (English, with French summary). MR 778970
- V. G. Maz′ya, Prostranstva S. L. Soboleva, Leningrad. Univ., Leningrad, 1985 (Russian). MR 807364
- Robert J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), no. 2, 309–323. MR 1369395, DOI 10.1215/S0012-7094-95-08013-2
- Bruno Nazaret, Best constant in Sobolev trace inequalities on the half-space, Nonlinear Anal. 65 (2006), no. 10, 1977–1985. MR 2258478, DOI 10.1016/j.na.2005.05.069
- Alexander I. Nazarov, Dirichlet and Neumann problems to critical Emden-Fowler type equations, J. Global Optim. 40 (2008), no. 1-3, 289–303. MR 2373558, DOI 10.1007/s10898-007-9193-6
- A. I. Nazarov, Hardy-Sobolev inequalities in a cone, J. Math. Sci. (N.Y.) 132 (2006), no. 4, 419–427. Problems in mathematical analysis. No. 31. MR 2197336, DOI 10.1007/s10958-005-0508-1
- A. I. Nazarov, The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers, J. Math. Sci. (New York) 102 (2000), no. 5, 4473–4486. Function theory and applications. MR 1807067, DOI 10.1007/BF02672901
- Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. MR 547524
- G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
Bibliographic Information
- A. I. Nazarov
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
- MR Author ID: 228194
- Email: al.il.nazarov@gmail.com
- 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.
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 997-1006
- MSC (2010): Primary 46E35
- DOI: https://doi.org/10.1090/S1061-0022-2011-01180-X
- MathSciNet review: 2760091
Dedicated: Dedicated to V. M. Babich on the occasion of his 80th birthday