Annali di Matematica Pura ed Applicata

, Volume 124, Issue 1, pp 181–197 | Cite as

An estimate for the best constant in a Sobolev inequality involving three integral norms

  • Howard A. Levine
Article
Received:

Summary

Let H1(R2)denote the Sobolev space of all real valued functions on R2 which, together with their gradients, are square integrable. Let
$$\begin{gathered} \lambda \equiv \inf \iint\limits_{R^2 } {|\nabla \varphi |^2 dxdy(\iint\limits_{R^2 } {\varphi ^2 dxdy/}\iint\limits_{R^2 } {\varphi ^6 dxdy})^{1/2} \equiv \inf J_2 (\varphi )} \hfill \\ \varphi \in H^1 (R^2 ),\varphi \ne 0 \hfill \\ \end{gathered} $$
1weakly convergent subsequence and that if at least one weak limit is nonzero then there is a function ϕ0 εH1(R2)such that J2(ϕ0)=λand that λ∼π4/3.We provide as asymptotic formula for ϕ0namely, we show ϕ0(r)≈K0(r)· ·[K 0 4 (r)+1]−1/2as r→+∞,where K0 is the modified Bessel function of the second kind. An extension to the more general inequality
$$C = \inf (\int\limits_{R^m } {|\nabla \varphi |^p dx} )^{\alpha /p} (\int\limits_{R^m } {|\varphi |^q dx} )^{\beta /q} (\int\limits_{R^m } {|\varphi |^r dx} )^{ - 1/r} $$
, β>0,α+β= 1and r−1=βq−1+α(p−1−m−1),p, q, r>1is briefly discussed.

Keywords

Sobolev Space Bessel Function Asymptotic Formula Sobolev Inequality Weak Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. Aubin,Problèmes Isopérimétriques et espaces de Sobolev, J. Diff. Geom.,11 (1976), pp. 573–598.Google Scholar
  2. [2]
    G. A. Bliss,An integral inequality, J. Lond. Math. Soc.,5 (1930), pp. 40–46.Google Scholar
  3. [3]
    H. Federer,Curvature measure, Trans. Am. Math. Soc.,93 (1959), pp. 418–490.Google Scholar
  4. [4]
    W. Fleming -R. Rishel,An integral formula for total gradient variation, Arch. der Math.,11 (1960), pp. 218–220.Google Scholar
  5. [5]
    A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.Google Scholar
  6. [6]
    K. Hilden,Symmetrization of functions in Sobolev space and the isoperimetric inequality Man. Math.,18 (1976), pp. 215–235.Google Scholar
  7. [7]
    E. Kamke,Differentialgleichungen, Lösungsmethoden und Lösungen, Mathematik und Ihre Anwendung, Band 18, Akad. Verlag., Leipzig, 1943.Google Scholar
  8. [8]
    L. E. Payne,Uniqueness criteria for steady state solutions of the Navier Stokes equations, Simpos. Internaz. Appl. Anal. Fis. Mat. (Cagliari-Sassari, 1964), Edizioni Cremonese, Roma (1965), pp. 130–153.Google Scholar
  9. [9]
    L. E. Payne,Uniqueness and continuous dependence criteria for the Navier-Stokes equations, Rocky Mountian J. Math.,2 (1971), pp. 641–660.Google Scholar
  10. [10]
    G.Polya,Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zussammenhängende Gebiete, S. Ber. Preuss. Akad. Wiss. (1928), pp. 228–232 and 280–282.Google Scholar
  11. [11]
    G.Rosen,Sobolev type lower bounds on ∥∇ ψ∥2for arbitrary regions in two dimensional Euclidean space, Quart. Appl. Math. (1976), pp. 200–202.Google Scholar
  12. [12]
    J. Serrin,Mathematical principles of classical fluid mechnaics, Handbuch der Physik, Band 8/1, Strömungsmechanik I, Springer V, Berlin, 1959.Google Scholar
  13. [13]
    J. Serrin,On the stability of viscous fluid motions, Arch. Rat. Mech. Anal.,3 (1959), pp. 1–13.Google Scholar
  14. [14]
    W. Spiegel,Über die Symmetrisierung stetiger Funktionen im euclidischen Raum, Arch. der Math.,24 (1973), pp. 545–551.Google Scholar
  15. [15]
    E. Sperner,Symmetrisierung fur Funtionen mehrer reeller Variablen, Man. Math.,18 (1976), pp. 215–235.Google Scholar
  16. [16]
    E. Sperner,Zur Symmetrisierung von Funktionen auf Sphären, Math. Z.,134 (1973), pp. 317–372.Google Scholar
  17. [17]
    B. Sz. Nagy,Integral inequalities between a function and its derivative, Acta Univ. Szeged,10 (1941), pp. 64–74 (German).Google Scholar
  18. [18]
    G. Talenti,Best constant in Sobolev inequality, Annali di Mat. pura. ed Applicata,110 (1976), pp. 353–372.Google Scholar
  19. [19]
    G. N.Watson,A Treatise on the Theory of Bessel Functions, Camb. U. Press 1922.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1980

Authors and Affiliations

  • Howard A. Levine
    • 1
  1. 1.AmesIowa

Personalised recommendations