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ISSN 1088-6850(e) ISSN 0002-9947(p)

Extremal functions for Moser's inequality

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

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