Brézis-Wainger Inequality on Riemannian Manifolds
© Przemysław Górka. 2008
Received: 10 November 2007
Accepted: 29 April 2008
Published: 8 May 2008
The Brézis-Wainger inequality on a compact Riemannian manifold without boundary is shown. For this purpose, the Moser-Trudinger inequality and the Sobolev embedding theorem are applied.
There is no doubt that the Brézis-Wainger inequality (see ) is a very useful tool in the examination of partial differential equations. Namely, a lot of estimates to a solution of PDE are obtained with the help of the Brézis-Wainger inequality. Especially, the inequality is often applied in the theory of wave maps.
In this paper, we extend the Brézis-Wainger result onto a compact Riemannian manifold. We show the following theorem.
where is a positive constant.
The proof relies on the application of a Moser-Trudinger inequality (see Theorem 2.2) and the Sobolev embedding theorem (see Theorem 2.1). Moreover, we will use the integral representation of a smooth function via the Green function (see ).
In order to make this paper more readable, we recall some definitions and facts from the theory of Sobolev spaces on Riemannian manifolds. In particular, we present useful inequalities and embeddings.
where by we have denoted the Riemannian measure on the manifold .
We define the Sobolev space as a completion of with respect to .
We close this section stating the following results, which will be used in the proof of the main result.
Let be a smooth, compact Riemannian -manifold. Then, for any real numbers and any integers , if , then . Moreover, there exists a constant such that for all , the following inequality holds:
Theorem 2.2 (Moser-Trudinger Inequality ).
Let be a smooth, compact Riemannian -manifold and a positive integer, strictly smaller than . There exist a constant and such that for all with and , the following inequality holds:
3. Proof of the Main Result
In this section, we will prove the main result, that is, Theorem 1.1.
Let us notice that by the assumptions we have an embedding . First of all, we show the following lemma.
where is a constant from the Moser-Trudinger inequality (see Theorem 2.2).
From this, the proof of Lemma 3.1 follows.
Let us recall (see ) that for , a compact Riemannian -manifold, there exists a Green function such that
where is a Riemannian volume of the manifold , and is the Laplace-Beltrami operator on a manifold;
and is a Riemannian distance from to .
and we have that .
Using elementary calculations, one can easily show the lemma.
where is the exponent from the Sobolev theorem.
This finishes the proof of the inequality in the case such that and . Subsequently, one can easily obtain the inequality for such that .
Finally, we take . This completes the proof.
The author wishes to thank Professor Yuxiang Li for pointing out the paper . Moreover, the author thanks the referees for comments and invaluable suggestions.
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