Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extremal functions for Moser's inequality

Author: Kai-Ching Lin
Journal: Trans. Amer. Math. Soc. 348 (1996), 2663-2671
MSC (1991): Primary 49J10
MathSciNet review: 1333394
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega $ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega $. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that

\begin{equation*}% \frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx \leq c_{o}% , \end{equation*}

where $|\Omega |$ is the Lebesgue measure of $\Omega $, and $\omega _{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the

\begin{equation*}% \sup \{\frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 \} % \end{equation*}

is attained. Earlier results include Carleson-Chang (1986, $\Omega $ is a ball in any dimension) and Flucher (1992, $\Omega $ is any domain in 2-dimensions).

References [Enhancements On Off] (What's this?)

  • 1. Adimurthi, Y. Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl Sci. (4) 17 (1990), 393-414. MR 91j:35016
  • 2. Bandle, C., Flucher, M., Harmonic radius and concentartion of energy, hyperbolic radius and Liouville's equations $\triangle u = e^{u}$ and $\triangle u = u^{\frac {n + 2}{n -2}}$, to appear in Siam Review.
  • 3. Carleson, L., and Chang, S.-Y. A., On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. Astro. (2) 110 (1986), 113-127. MR 88f:46070
  • 4. Federer, H., Geometric measure theory, Springer-Verlag (1969). MR 41:1976
  • 5. Flucher, M., Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comm. Math. Helv. 67 (1992), 471-497. MR 93k:58073
  • 6. Heinonen J., Kilpelainen, T, and Martio, O., Non-linear potential theory of degenerate elliptic equations Oxford Sci. Pub. (1993). MR 94e:31003
  • 7. Kawohl, B., Reaarangements and convexity of level sets in PDE, Lecture Notes in Math. 1150 (1985). MR 87a:35001
  • 8. Kichenassamy, S., and Veron L., Singular solutions of the p-Laplace equation, Math. Ann. 275 (1986), 599-615. MR 87j:35096
  • 9. Kilpelainen, T., and Maly, J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 19 (1992), 591-613. MR 94c:35091
  • 10. Lin, K., Moser's inequality and the n-Laplacian, to appear.
  • 11. Lions, P. L., The concentration-compactness principle in the calculus of variation, the limit case, Part I, Rev. Mat. Iberoamericana 1 (1985), 145-201. MR 87j:49012
  • 12. Moser, J., A sharp form of an inequality by N. Trudinger, Indianna Univ. Math. J. 20 (1971), 1077-1092. MR 46:662
  • 13. Struwe, M., Critical points of embeddings of $H_{0}^{1, n}$ into Orlicz spaces, Ann. Inst. H. Poincare Anal. Non Lineaire, Vol. 5 (1988), 425-464. MR 90c:35084
  • 14. Trudinger, N. S., On embeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-484. MR 35:7121

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 49J10

Retrieve articles in all journals with MSC (1991): 49J10

Additional Information

Kai-Ching Lin
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487

Received by editor(s): January 25, 1995
Received by editor(s) in revised form: May 30, 1995
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society