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Brézis-Wainger Inequality on Riemannian Manifolds

Open Access
Research Article

Abstract

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.

Keywords

Partial Differential Equation Riemannian Manifold Sobolev Space Green Function Integral Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

There is no doubt that the Brézis-Wainger inequality (see [1]) 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.

Theorem 1.1.

where Open image in new window 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 [2]).

2. Preliminaries

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.

Let Open image in new window be a smooth, compact Riemannian manifold without boundary. We will denote by Open image in new window a space of smooth real functions. For Open image in new window and integer Open image in new window , we denote by Open image in new window the m th covariant derivative of Open image in new window . Next, for Open image in new window and for a fixed integer Open image in new window and a real Open image in new window , we set

where by Open image in new window we have denoted the Riemannian measure on the manifold Open image in new window .

We define the Sobolev space Open image in new window as a completion of Open image in new window with respect to Open image in new window .

We close this section stating the following results, which will be used in the proof of the main result.

Theorem 2.1 (Sobolev Embedding Theorem [3, 4]).

Let Open image in new window be a smooth, compact Riemannian Open image in new window -manifold. Then, for any real numbers Open image in new window and any integers Open image in new window , if Open image in new window , then Open image in new window . Moreover, there exists a constant Open image in new window such that for all Open image in new window , the following inequality holds:

Theorem 2.2 (Moser-Trudinger Inequality [5]).

Let Open image in new window be a smooth, compact Riemannian Open image in new window -manifold and Open image in new window a positive integer, strictly smaller than Open image in new window . There exist a constant Open image in new window and Open image in new window such that for all Open image in new window with Open image in new window and Open image in new window , the following inequality holds:

Let us stress that this inequality is a generalization of the Moser and Trudinger result (see [6, 7, 8]).

3. Proof of the Main Result

In this section, we will prove the main result, that is, Theorem 1.1.

Proof.

Let us notice that by the assumptions we have an embedding Open image in new window . First of all, we show the following lemma.

Lemma 3.1.

Proof.

Let us put
in Young's inequality
where Open image in new window . Then, we obtain an estimate

where Open image in new window is a constant from the Moser-Trudinger inequality (see Theorem 2.2).

Next, we can estimate
Subsequently, we can apply the Moser-Trudinger inequality to the right-hand side of the above inequality:

From this, the proof of Lemma 3.1 follows.

Now, we can go back to the proof of Theorem 1.1. First, we prove the theorem for Open image in new window such that Open image in new window and Open image in new window . Replacing Open image in new window by Open image in new window if necessary, we may suppose that

for Open image in new window .

Let us recall (see [2]) that for Open image in new window , a compact Riemannian Open image in new window -manifold, there exists a Green function Open image in new window such that

where Open image in new window is a Riemannian volume of the manifold Open image in new window , and Open image in new window is the Laplace-Beltrami operator on a manifold;

(2)there exists a constant Open image in new window such that for any Open image in new window ,

and Open image in new window is a Riemannian distance from Open image in new window to Open image in new window .

Let us define the map Open image in new window . Next, we apply the first property of the Green function to the map Open image in new window and to the point Open image in new window . Namely,
Subsequently, by the second property of the Green function we can estimate Open image in new window as follows:
Now, we will try to estimate the last term in the inequality (3.12). Let us notice that if Open image in new window , then Open image in new window . Next, by the Sobolev theorem (see Theorem 2.1),

and we have that Open image in new window .

Using elementary calculations, one can easily show the lemma.

Lemma 3.2.

There exist Open image in new window and a finite Open image in new window such that the following equality holds:

where Open image in new window is the exponent from the Sobolev theorem.

By Hölder's inequality with exponents Open image in new window , we can estimate the inequality (3.12) as follows:
where Open image in new window and Open image in new window is the exponent from Lemma 3.2. Finally, by Lemma 3.1 and the Sobolev theorem, we obtain
Let us stress that since Open image in new window , the constant Open image in new window does not depend on Open image in new window . We can rewrite the inequality (3.16) as follows:
Taking into account Open image in new window , we obtain

This finishes the proof of the inequality in the case Open image in new window such that Open image in new window and Open image in new window . Subsequently, one can easily obtain the inequality for Open image in new window such that Open image in new window .

Now, we prove the theorem for an arbitrary Open image in new window such that Open image in new window . We apply the density argument. Namely, for any Open image in new window , there exists Open image in new window such that
Next, we define
Such Open image in new window has zero-mean value. Moreover, the following inequality holds:

Finally, we take Open image in new window . This completes the proof.

Notes

Acknowledgments

The author wishes to thank Professor Yuxiang Li for pointing out the paper [5]. Moreover, the author thanks the referees for comments and invaluable suggestions.

References

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Copyright information

© Przemysław Górka. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics and Information SciencesWarsaw University of TechnologyWarsawPoland

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