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.
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.
We close this section stating the following results, which will be used in the proof of the main result.
Theorem 2.2 (Moser-Trudinger Inequality ).
3. Proof of the Main Result
In this section, we will prove the main result, that is, Theorem 1.1.
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
Using elementary calculations, one can easily show the lemma.
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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