Skip to main content
SpringerLink
Log in

Extremal functions for the trudinger-moser inequality in 2 dimensions

  • Published:
Commentarii Mathematici Helvetici

Abstract

We prove that theTrudinger-Moser constant

$$\sup \left\{ {\int_\Omega {\exp (4\pi u^2 )dx:u \in H_0^{1,2} (\Omega )\int_\Omega {\left| {\nabla u} \right|^2 dx \leqslant 1} } } \right\}$$

is attained on every 2-dimensional domain. For disks this result, is due to Carleson-Chang. For other domains we derived an isoperimetric inequality which relates the ratio of the supremum of, the functional and its maximal limit on concentrating sequences to the corresponding quantity for disks. A conformal rearrangement is introduced to prove this inequality.

I would like to thank Jürgen Moser and Michael Struwe for helpful advice and criticism.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adimurthi.Summary of the results on critical exponent problem in R 2, preprint (1990).

  2. Adimurthi.Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Vol.17, 3 (1990), 393–414.

    MathSciNet  Google Scholar 

  3. Bahri A., Coron J. M..On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math.41 (1988), 253–294.

    MathSciNet  Google Scholar 

  4. Carleson L., Chang S. A..On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. Astro. (2)110 (1986), 113–127.

    MathSciNet  Google Scholar 

  5. Federer, H. Geometric measure theory, Springer-Verlag (1969).

  6. Han Z-C..Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, Vol.8, No 2 (1991), 159–174.

    Google Scholar 

  7. Kawohl B. Rearrangements and convexity of level sets in PDE, Lecture notes in Math.1150 (1985).

  8. Lions P. L. The concentration compactness principle in the calculus of variations, The limit case, Part 1, Rev. Mat. Iberoamericana1 (1985).

  9. Moser, J..A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., Vol.20, No. 11 (1971), 1077–1092.

    Article  Google Scholar 

  10. Pohozaev S.. Eigenfunctions of the equationΔu+λf(u)=0, Soviet Math. Dokl.6 (1965), 1408–1411.

    Google Scholar 

  11. Pólya G., Szegö G. Isoperimetric inequalities in mathematical physics, Princeton University Press (1951).

  12. Rey O. The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Vol.89, No. 1 (1990).

  13. Schoen R. M. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Lecture Notes in Math. No. 1365 (1989).

  14. Struwe M..Critical points of embedding of H o 1,2 into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, Vol.5, No. 5 (1988), 425–464.

    MathSciNet  Google Scholar 

  15. Trudinger N. S..On embeddings into Orlicz spaces and some applications, J. Math. Mech.17 (1967), 473–484.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ETH-Zürich HG G 36.1, Rämistr. 101, CH-8092, Zürich, Switzerland

    Martin Flucher

Authors
  1. Martin Flucher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flucher, M. Extremal functions for the trudinger-moser inequality in 2 dimensions. Commentarii Mathematici Helvetici 67, 471–497 (1992). https://doi.org/10.1007/BF02566514

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02566514

Keywords

Search

Navigation