Abstract
We prove that theTrudinger-Moser constant
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.
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References
Adimurthi.Summary of the results on critical exponent problem in R 2, preprint (1990).
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.
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.
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.
Federer, H. Geometric measure theory, Springer-Verlag (1969).
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.
Kawohl B. Rearrangements and convexity of level sets in PDE, Lecture notes in Math.1150 (1985).
Lions P. L. The concentration compactness principle in the calculus of variations, The limit case, Part 1, Rev. Mat. Iberoamericana1 (1985).
Moser, J..A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., Vol.20, No. 11 (1971), 1077–1092.
Pohozaev S.. Eigenfunctions of the equationΔu+λf(u)=0, Soviet Math. Dokl.6 (1965), 1408–1411.
Pólya G., Szegö G. Isoperimetric inequalities in mathematical physics, Princeton University Press (1951).
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).
Schoen R. M. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Lecture Notes in Math. No. 1365 (1989).
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.
Trudinger N. S..On embeddings into Orlicz spaces and some applications, J. Math. Mech.17 (1967), 473–484.
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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
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DOI: https://doi.org/10.1007/BF02566514