Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 475957, 15 pages
doi:10.1155/2008/475957

Copyright © 2008 Florica C. Cîrstea and Sever S. Dragomir. 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.

Abstract

We use the potential theory to give integral representations of functions in the Sobolev spaces W1,p(Ω), where p≥1 and Ω is a smooth bounded domain in ℝN (N≥2). As a byproduct, we obtain sharp inequalities of Ostrowski type.

1. Introduction and Main Results

Let and let denote the canonical inner product on . If stands for the area of the surface of the -dimensional unit sphere, then , where is the gamma function defined by for (see [1, Proposition 0.7]).

Let denote the normalized fundamental solution of Laplace equation: (1.1)

Unless otherwise stated, we assume throughout that is a bounded domain with boundary . Let denote the unit outward normal to and let indicate the -dimensional area element in . The Green-Riemann formula says that any function satisfying can be represented in as follows (see [2, Section 2.4]): (1.2) where is the normal derivative of at . In particular, if (the set of functions in with compact support in ), then (1.2) leads to the representation formula (1.3)

For a continuous function on , the double-layer potential with moment is defined by (1.4)

Expression (1.4) may be interpreted as the potential produced by dipoles located on ; the direction of which at any point coincides with that of the exterior normal , while its intensity is equal to . The double-layer potential is well defined in and it satisfies the Laplace equation in (see Proposition 2.8). For other properties of the double-layer potential, see Lemma 2.9 and Proposition 2.10.

The double-layer potential plays an important role in solving boundary value problems of elliptic equations. The representation of the solution of the interior/exterior Dirichlet problem for Laplace's equation is sought as a double-layer potential with unknown density . An application of property (2.14) leads to a Fredholm equation of the second kind on in order to determine the function (see, e.g., [3]).

In many problems of mathematical physics and variational calculus, it is not sufficient to deal with classical solutions of differential equations. One needs to introduce the notion of weak derivatives and to work in Sobolev spaces, which have become an indispensable tool in the study of partial differential equations.

For , we denote by the Sobolev space defined by(1.5) For , we define and write . The Sobolev space is endowed with the norm (1.6) where stands for the usual norm on . The closure of in the norm of is denoted by . For details on Sobolev spaces, we refer to [2, 4], or [5].

Since is bounded, we have for every .

The following representation holds for functions in with (see Remark 2.3): (1.7)

We first give an integral representation of functions in for any .

Theorem 1.1. For any with , there is a sequence in such that (1.8) (1.9)

Remark 1.2. If , then there exists a sequence in for which (1.8) holds. Thus, we regain (1.7) for any function in .

Under a suitable smoothness condition, the representation of Theorem 1.1 can be refined for functions in with (see Theorem 1.3). Using Morrey's inequality, one can prove that functions in the Sobolev space with are classically differentiable almost everywhere in (cf. [2, page 176] or [4]). By Proposition 2.13, any function in with is uniformly Hölder continuous in with exponent (after possibly being redefined on a set of measures ). In particular, any function in with is continuous on , and thus it has a well-defined trace which is bounded.

The proof of Theorem 1.1 relies on the density of in as well as the following result.

Theorem 1.3. Assume that , where and is a finite family of points in .

(a) If , then can be represented as follows: (1.10)

(b) If and , then (1.11)

Remark 1.4. (i) If on , then Theorem 1.3 recovers Gauss formula (see Lemma 2.9).

(ii) Theorem 1.3 leads to the mean value theorems for harmonic functions (see Remark 5.4).

(iii) If such that , then by combining Theorem 1.3 and Proposition 2.7, we regain the Green-Riemann representation formula (1.2).

This paper is organized as follows. In Section 2, we include some known results that are necessary later in the paper. Section 3 is dedicated to the proof of Theorem 1.3. Based on it, we prove Theorem 1.1 in Section 4. We conclude the paper with a representation of smooth functions in with in terms of the integral mean value over the domain (see Theorem 5.1 in Section 5). As a byproduct of our main results, we obtain a sharp estimate of the difference between the value of a function and the double-layer potential with moment .

2. Preliminaries

Lemma 2.1 (see [4, Theorem IV.9]). Let be an open set. Let be a sequence in , , and let be such that .

Then, there exist a subsequence and a function such that

(a) a.e. in ,

(b) for all , a.e. in .

For fixed , we define the operator by (2.1)

Lemma 2.2. (i) If , then the operator is compact.

(ii) If , then the operator is compact.

Remark 2.3. If is a bounded domain and with , then (1.7) holds. Indeed, given by (1.1) has weak derivatives and for every . If , then by the definition of weak derivatives, we have (2.2) Thus, using (1.3), we find (1.7) for every . Now, if , we take a sequence in such that in as . Thus, for each with , we have (2.3) By Lemma 2.2, each operator is compact from to . Thus, in as implies that in as . By Lemma 2.1, we have (up to a subsequence of ) () () and a.e. (since in as ). By passing to the limit in (2.3), we conclude (1.7).

Lemma 2.4 (see [5, Lemma 5.47]). Let and let be a domain of finite volume in .

If , then (2.4) where the constant depends on and but not on or .

By a vector field, we understand an -valued function on a subset of . If is a differentiable vector field on an open set , the divergence of on is defined by (2.5)

Proposition 2.5 (the divergence theorem). If is a bounded domain with boundary and is a vector field of class , then (2.6)

If is a domain to which the divergence theorem applies, then we have the following.

Proposition 2.6 (Green's first identity). If , then the following holds: (2.7)

Proposition 2.7. Let be a bounded domain with boundary. If such that , then for every , one has (2.8)

Proof. If , then (2.8) follows from Proposition 2.6 (since belongs to ). For fixed, we choose such that , where denotes the open ball of radius centered at . By Proposition 2.6 (applied on ), we find (2.9)

Since and , we have that and are integrable on . We see that (2.10) Indeed, for some constant , we have (2.11) Thus, passing to the limit in (2.9) and using (2.10), we obtain (2.8).

We next give some properties of the double-layer potential defined by (1.4) (see [1]).

Proposition 2.8. If is a continuous function on , then

(i) given by (1.4) is well defined for all , (ii) for all .

Lemma 2.9 (Gauss lemma). Let be the double-layer potential with moment , that is, (2.12) Then, one has (2.13)

Proposition 2.10. If is continuous on and , then (2.14)

Remark 2.11. If , then , for each .

Indeed, by Propositions 2.8 and 2.10, the function defined by for and for is continuous on . It follows that and . But on so that . Thus, for each , we have (2.15) which shows that .

Definition 2.12. A Lipschitz domain (or domain with Lipschitz boundary) is a domain in whose boundary can be locally represented as the graph of a Lipschitz continuous function.

Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. All smooth and many piecewise smooth boundaries are Lipschitz boundaries.

Proposition 2.13 (see [2.13, Theorem 7.2.136]). Let be a Lipschitz domain in . If , then is continuously embedded in with .

Proposition 2.14 (see [2.14, page 155]). If is a Lipschitz domain, then is dense in for .

3. Proof of Theorem 1.3

Since is bounded, we can assume without loss of generality that .

Proof of (a). Suppose that . Then, with (cf. Proposition 2.13).

Proof of (1.10) When . We define as follows: (3.1)Note that . We overcome this problem by choosing small enough such that , respectively, (), is contained within and every two such closed balls are disjoint. Therefore, , where .

Using Proposition 2.5, we arrive at (3.2) We see that (3.3) Indeed, by Proposition 2.13, there exists a constant such that (3.4) Notice that, for each with , there exists a constant such that (3.5) (since ). Hence, if such that , then (3.6) By (3.2)–(3.6) and Gauss lemma, it follows that (3.7) Recall that is harmonic on . Thus, from (3.1), we derive that (3.8) From Lemma 2.2(ii), we know that (3.9) From (3.7) and (3.8), we find (3.10) which concludes the proof of (1.10) for .

Proof of (1.10) When . We apply (1.10) to get with . Then, let . Thus, using (3.9) and the continuity of on , we obtain (3.11) From Proposition 2.10, we know that (3.12) By combining (3.11) and (3.12), we attain (1.10).

Proof of (b). Assume that and . Let be fixed.

We define the vector field by (3.13) Clearly, . Let be fixed such that for every and for all with . Set . By applying Proposition 2.5 to , we obtain (3.14) If , then for every (since ). Hence, for each , (3.15) By (3.14) and (3.15), it follows that (3.16) Note that is continuous on . By Hölder's inequality, is integrable on . Since is harmonic on , we find (3.17) Therefore, using (3.16), we obtain (3.18) This completes the proof of Theorem 1.3.

4. Proof of Theorem 1.1

As before, we can assume that with . By Proposition 2.14, there exists a sequence such that in , that is, (4.1) From Lemma 2.1, we know that, up to a subsequence (relabeled ), (4.2) Since for every , we can apply Theorem 1.3 to each and obtain (4.3) Using the definition of in (2.1), we write (4.4) From (4.1) and Lemma 2.2, it follows that for every , (4.5) Hence, passing eventually to a subsequence (denoted again by ), we have (4.6) This, jointly with (4.4), implies that (4.7) Hence, passing to the limit in (4.3) and using (4.2), we reach (1.8).

Proof of (1.9). Let be arbitrary. Then, is continuous on . Let denote the conjugate exponent to (i.e., ). By Hölder's inequality, (4.8) Thus, using (4.1) and Lemma 2.4, we infer that (4.9) Letting in (4.3), we conclude (1.9). This finishes the proof of Theorem 1.1.

5. Other Results and Applications to Inequalities

If is absolutely continuous on , then the Montgomery identity holds: (5.1) where is given by (5.2) In the last decade, many authors (see, e.g., [6] and the references therein) have extended the above result for different classes of functions defined on a compact interval, including functions of bounded variation, monotonic functions, convex functions, -time differentiable functions whose derivatives are absolutely continuous or satisfy different convexity properties, and so forth, and they pointed out sharp inequalities for the absolute value of the difference (5.3) The obtained results have been applied in approximation theory, numerical integration, information theory, and other related domains.

If is absolutely continuous on , then we have the following Ostrowski-type inequalities (see, e.g., [6, page 2]): (5.4) where is the conjugate exponent to . The constants , , and are best possible in the sense that they cannot be replaced by smaller constants.

If the function has continuous partial derivatives , , and on , then one has the representation (see [6, page 307]) (5.5) for each where is defined by (5.2) and is the corresponding kernel for the interval Another representation for is (5.6) for each provided is continuous on (see [6, page 294]).

For various Ostrowski-type inequalities, the reader is referred to the book in [6, Chapters 5 and 6] and the papers in [7, 8].

In this section, we give a representation formula for in terms of the integral mean value over (under the same assumptions on as in Theorem 1.3).

Theorem 5.1. One assumes that , where and is a finite family of points in . The following representation formula holds: (5.7)

Proof. We prove that (5.8) Let be arbitrary. We define by . Let be small such that for every and for all with . Set . We have . By Proposition 2.5, we find (5.9) For , we choose large such that , for every . Hence, (5.10) which implies that (5.11) Obviously, and is integrable on . Therefore, we have (5.12) Passing to the limit in (5.9), then using (5.11) and (5.12), we reach (5.8).

Using representation (1.10) of with and representation (5.8) with , we conclude (5.7).

Remark 5.2. More generally, in the framework of Theorem 5.1, one has (5.13)

As a consequence of Theorems 1.3 and 5.1, we obtain the following.

Corollary 5.3. Assume that , where and is a finite family of points in . The following hold.

(i) An arbitrary value of is compared below with the double-layer potential with moment : (5.14) where denotes the conjugate coefficient of (i.e., ). Moreover, for fixed, the equality in (5.14) is established for the nontrivial function if , respectively, with if .

(ii) For each and such that , one has (5.15)

In addition, (5.16) where the equality is achieved for if and if .

Proof. (i) From with , we have so that the right-hand side of (5.14) is finite (see Lemma 2.4). By (1.10) and Hölder's inequality, we have (5.17) Let be fixed. We define by if and if . Clearly, we have , and for every , (5.18) Since , we infer that and (5.19) By (1.10) and (5.18), the left-hand side (LHS) of (5.14) for is (5.20) A simple calculation shows that the right-hand side of (5.14) for equals the above LHS.

(ii) The first identity of (5.15) follows from Theorem 5.1, while the second follows from Theorem 1.3 (with and ). Notice that (5.21) By applying (5.14) with and , we find (5.16).

Remark 5.4. Corollary 5.3(ii) leads to the mean value theorems for harmonic functions. Indeed, if is harmonic on , then for every ball with , we have (5.22) This, jointly with (5.15), implies that (5.23)

Acknowledgment

The authors thank the referees for the useful comments on the first version of this paper.

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