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Asymptotic expansions and extremals for the critical Sobolev and Gagliardo–Nirenberg inequalities on a torus

Published online by Cambridge University Press:  22 May 2013

Michele Bartuccelli ,
Jonathan Deane  and
Sergey Zelik
Michele Bartuccelli
Affiliation:
University of Surrey, Department of Mathematics, Guildford GU2 7XH, UK (m.bartuccelli@surrey.ac.uk; j.deane@surrey.ac.uk; s.zelik@surrey.ac.uk)
Jonathan Deane
Affiliation:
University of Surrey, Department of Mathematics, Guildford GU2 7XH, UK (m.bartuccelli@surrey.ac.uk; j.deane@surrey.ac.uk; s.zelik@surrey.ac.uk)
Sergey Zelik
Affiliation:
University of Surrey, Department of Mathematics, Guildford GU2 7XH, UK (m.bartuccelli@surrey.ac.uk; j.deane@surrey.ac.uk; s.zelik@surrey.ac.uk)
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Abstract

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We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 3 , June 2013 , pp. 445 - 482
Copyright
Copyright © Royal Society of Edinburgh 2013