Skip to main content

An inequality between u* and |grad u|*

  • Chapter
General Inequalities 6

Abstract

We prove an inequality between the decreasing rearrangement of a function u and the decreasing rearrangement of the length of the gradient of u.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.A. Adams, Sobolev spaces. Academic Press 1975.

    Google Scholar 

  2. A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz. Boll. Unione Mat. It. (5) 14A (1977), 148–156.

    Google Scholar 

  3. A. Alvino, P.-L. Lions and G. Trombetti, On optimization problems with prescribed rearrangements. Nonlinear Analysis TMA 13 (1989), 185–220.

    Article  Google Scholar 

  4. G. Aronsson and G. Talenti, Estimating the integral of a function in terms of a distribution function of its gradient. Boll. Unione Mat. It. (5) 18B (1981), 885–894.

    Google Scholar 

  5. T. Aubin, Problèmes isopérimetriques et espaces de Sobolev. J. Diff. Geometry 11 (1976), 573–598.

    Google Scholar 

  6. J. Brothers and W.P. Ziemer, Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384 (1988), 153–179.

    Google Scholar 

  7. E. De Giorgi, Sulla proprietà isoperimetrica dell’ipersfera. Mem. Acc. Naz. Lincei (8) 5 (1958), 33–44.

    Google Scholar 

  8. W. Fleming and R. Rishel, An integral formula for total gradient variation. Arch. Math. 11 (1960), 218–222.

    Article  Google Scholar 

  9. J. J. F. Fournier, Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality. Ann. Mat. Pura Appl. (4) 148 (1987), 53–76.

    Google Scholar 

  10. E. Giarusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations. Nonlinear Analysis TMA 8 (1984), 289–299.

    Article  Google Scholar 

  11. E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser 1984.

    Google Scholar 

  12. G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge Univ. Press 1934.

    Google Scholar 

  13. A. Hunt, On L(p, q) spaces. Enseignement math. 12 (1966), 249–276.

    Google Scholar 

  14. B. Kawohl, Rearrangements and convexity of level sets in PDE. Lecture Notes in Math., Vol. 1150, Springer-Verlag 1985.

    Google Scholar 

  15. H.A. Levine, An estimate for the best constant in a Sobolev inequality involving three integral norms. Ann. Mat. Pura Appl. 124 (1980), 181–197.

    Article  Google Scholar 

  16. E.H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118 (1983), 349–374.

    Google Scholar 

  17. V.G. Maz’ja, Sobolev spaces. Springer-Verlag 1985.

    Google Scholar 

  18. J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Math. J. 20 (1971), 1077–1092.

    Article  Google Scholar 

  19. G.O. Okikiolu, Aspects of the theory of bounded integral operators in LP-spaces. Academic Press 1971.

    Google Scholar 

  20. R. O’Neil, Convolution operators and L(p, q) spaces. Duke Math. J. 30 (1963), 129–142.

    Google Scholar 

  21. S. Poornima, An embedding theorem for the Sobolev space W’-’. Bull. Sci. Math. 107 (1983), 253–259.

    Google Scholar 

  22. S.L. Sobolev, On a theorem of functional analysis. Math. Sb. 4 (1938), 471–497.

    Google Scholar 

  23. E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press 1971.

    Google Scholar 

  24. G. Talenti, Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.

    Google Scholar 

  25. G. Talenti, The standard isoperimetric theorem. Handbook of convex geometry( P. Gruber and J.M. Wills editors), forthcoming.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Basel AG

About this chapter

Cite this chapter

Talenti, G. (1992). An inequality between u* and |grad u|*. In: Walter, W. (eds) General Inequalities 6. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 103. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7565-3_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7565-3_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7567-7

  • Online ISBN: 978-3-0348-7565-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics