Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011, Article ID 901084, 12 pages
http://dx.doi.org/10.1155/2011/901084
Research Article

-Distributions: An Extension of -Measures to an Setting

1Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička c. 30, 10 000 Zagreb, Croatia
2Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81 000 Podgorica, Montenegro

Received 13 March 2011; Revised 22 May 2011; Accepted 20 June 2011

Academic Editor: P. J. Y. Wong

Copyright © 2011 Nenad Antonić and Darko Mitrović. 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 continuity of Fourier multiplier operators on to introduce the -distributions—an extension of -measures in the framework. We apply the -distributions to obtain an version of the localisation principle and reprove the Murat - variant of div-curl lemma.

1. Introduction

In the study of partial differential equations, quite often it is of interest to determine whether some weakly convergent sequence converges strongly. Various techniques and tools have been developed for that purpose (for the state of the art twenty years ago, see [1]); of more modern ones, we only mention the -measure of Tartar [2], independently introduced by Gérard [3] under the name of microlocal defect measures. -measures proved to be very powerful tool in a number of applications (see, e.g., [413] and references therein, which is surely an incomplete list). The main theorem on the existence of -measures, in an equivalent form suitable for our purposes, reads as following:

Theorem 1.1. If scalar sequences weakly in , then there exist subsequences and a complex Radon measure on such that, for every and every , where is the Fourier multiplier operator with the symbol :

We call the measure the -measure corresponding to the sequence . In fact, it corresponds to the nondiagonal element of the corresponding matrix Radon measure of the vector function (cf. [14]).

Remark 1.2. After applying the Plancherel theorem, the term under the limit sign in Theorem 1.1 takes the form where by we denote the Fourier transform on (with the inverse ). Of course, is extended by homogeneity (i.e., ) to .
In the particular case of , roughly describes the loss of strong precompactness of sequence . Indeed, it is not difficult to see that if is strongly convergent in , then the corresponding -measure is trivial; on the other hand, if the -measure is trivial, then in (for the details in a similar situation see, [15]).

In order to explain how to apply this idea to -weakly converging sequences when , consider the integral in (1.1). The Cauchy-Schwartz inequality and the Plancherel theorem imply (see, e.g., [2, page 198]) where depends on a uniform bound for . In essence, this fact and the linearity of integral in (1.1) with respect to and enable us to state that the limit in (1.1) is a Radon measure (a bounded linear functional on ). Furthermore, the bound is obtained by a simple estimate and the fact that is a bounded sequence in .

In [3], the question whether it is possible to extend the notion of -measures (or microlocal defect measures in the terminology used there) to the framework is posed (see also [16, page 331]). We will consider only the case (i.e., ; its dual exponent we will consistently denote by ).

To answer that question, one necessarily needs precise bounds for the Fourier multiplier operator as a mapping from to . The bounds are given by the famous Hörmander-Mikhlin theorem [17, 18].

Definition 1.3. Let satisfy for some . Then, is called the Fourier multiplier on , if for any , and can be extended to a continuous mapping . One calls operator the -multiplier operator with symbol .

Theorem 1.4 (Hörmander-Mikhlin). Let have partial derivatives of order less than or equal to , where is the least integer strictly greater than (i.e., ). If, for some constant , then, for any and the associated multiplier operator , there exists a constant (depending only on the dimension ; see [18, page 367]) such that

Remark 1.5. It is important to notice that, according to [19, Section  3.2, Example  2], if the symbol of a multiplier is a function defined on the unit sphere , then the constant from Theorem 1.4 can be taken to be equal to .
By an application of Theorem 1.4, in Section 2 we are able to introduce -distributions (see Theorem 2.1)—an extension of -measures in the -setting. Its proof is the main result of the paper and forms Section 3. We conclude in Section 4 by an -variant of the localisation principle and a proof of an -variant of the div-curl lemma.

Remark 1.6. Recently, variants of -measures with a different scaling were introduced (the parabolic -measures [15, 20] and the ultraparabolic -measures [21]). We can apply the procedure from this paper to extend the notion of such -measures to the -setting in the same fashion as it is given here based on Theorem 1.1 for the classical -measures.

2. A Generalisation of -Measures

We have already seen (Remark 1.2) that an -measure corresponding to a sequence in can describe its loss of strong compactness. We would like to introduce a similar notion describing the loss (at least in ) of strong compactness for sequences weakly converging in .

Consider a sequence weakly converging to zero in and satisfying the following sequence of differential equations: where and strongly in the Sobolev space . When dealing with the latter equation, it is standard to multiply (2.1) by , for , where is the multiplier operator with symbol , , and then pass to the limit (see, e.g., [14, 22]). If , then we can apply the classical -measures to describe the defect of compactness for .

If we instead take , for , then we cannot apply the same tool. Here we propose the following replacement.

Theorem 2.1. If in and in for some , then there exist subsequences , and a complex valued distribution of order not more than in , such that, for every and one has: where is a multiplier operator with symbol .

We call the functional the -distribution corresponding to (a subsequence of) and . Of course, for , the weak convergence coincides with the weak convergence.

If we are given sequences and defined on an open set , then we can extend the functions by zero to , preserving the convergence, and then apply Theorem 2.1 in the above form. The resulting -distribution will be supported on , as it can easily be seen by taking test functions and supported within the complement of the closure .

Remark 2.2. Notice that, unlike what was the case with -measures, it is not possible to write (2.2) in a form similar to (1.3) since, according to the Hausdorff-Young inequality, only if . This means that we are not able to estimate , for , which would appear from (2.2) when rewriting it in a form similar to (1.3).

Remark 2.3. In Theorem 2.1, we clearly distinguish between and . For , and we can take ; in particular, this covers the classical case (including ). Even more, in this case , the assumptions of Theorem 2.1 imply that in and we can again use a classical framework, resulting in a distribution of order zero (a Radon measure, not necessary bounded), instead of a more general distribution of order . The real improvement in Theorem 2.1 is for the case .

Remark 2.4. For applications, it might be of interest to extend the result to vector-valued functions. In the case when and , the result is a matrix valued distribution , where and .
It should be noted that, in contrast to what is done with -measures, in general we cannot consider -distributions corresponding to the same sequence, but only to a pair of sequences, and -distribution would correspond to nondiagonal blocks for -measures [14] (see also the example at the beginning of Section 4).

3. Proof of Theorem 2.1

In order to prove the theorem, we need a consequence of Tartar's first commutation lemma [2, Lemma  1.7]. First, for and , define the Fourier multiplier operator and the operator of multiplication on , by the formulae Notice that satisfies the conditions of the Hörmander-Mikhlin theorem (see Remark 1.5). Therefore, and are bounded operators on , for any . We are interested in the properties of their commutator, .

Lemma 3.1. Let be bounded in both and , for some , and such that in the sense of distributions. Then the sequence strongly converges to zero in , for any .

Proof. If , then we can apply the classical interpolation inequality: for such that . As is a compact operator on by the first commutation lemma, while is bounded on , from (3.2) we get the claim.
In the case , notice that we do not have the boundedness of on , but only on , for . Therefore, we take and by the interpolation inequality conclude that is bounded in . Now, we can proceed as above, with replaced by .

Proof of Theorem 2.1. The first equality from (2.2) follows from the fact that the adjoint operator corresponding to is actually the multiplier operator (see [17, Theorem  7.4.3]). This means that (we take the duality product to be sesquilinear, i.e., antilinear in the second variable, in order to get the scalar product when ) which is exactly what we need. We can now concentrate our attention on the second equality in (2.2).
Since in , while for one has , according to the Hörmander-Mikhlin theorem for any and , it follows that
We can write , where form an increasing family of compact sets (e.g., closed balls around the origin of radius ); therefore for some . One has where is the characteristic function of . In the second equality, one has used Lemma 3.1.
This allows us to express the above integrals as bilinear functionals, after denoting :
Furthermore, is bounded by , as according to the Hölder inequality, Theorem 1.4 and Remark 1.5: where the constant depends on -norm and -norm of the sequences and , respectively.
For each , we can apply Lemma 3.2 (actually, the operators are defined in its proof) to obtain operators . Furthermore, for the construction of , we can start with a defining subsequence for , so that the convergence will remain valid on , in such a way obtaining that is an extension of .
This allows us to define the operator on : for, , we take such that , and set . Because of the above-mentioned extension property, this definition is good, and one has a bounded operator:
In such a way one has got a bounded linear operator on the space equipped with the uniform norm; the operator can be extended to its completion, the Banach space .
Now, we can define , which satisfies (2.2).
We can restrict to an operator defined only on ; as the topology on is stronger than the one inherited from , the restriction remains continuous. Furthermore, is the space of distributions of order , which is a subspace of . In such a way, one has a continuous operator from to , which by the Schwartz kernel theorem can be identified to a distribution from (for details cf. [23, Chapter VI]).

We conclude this section by a simple lemma and its proof, which was used in the proof of Theorem 2.1.

Lemma 3.2. Let and be separable Banach spaces and an equibounded sequence of bilinear forms on (more precisely, there is a constant such that, for each one has ).
Then, there exists a subsequence and a bilinear form (with the same bound ) such that

Proof. To each , we associate a bounded linear operator by The above expression clearly defines a function (i.e., is uniquely determined); it is linear in and bounded: Let be a countable dense subset; for each , the sequence is bounded in , so by the Banach theorem there is a subsequence such that By repeating this construction countably many times and then applying the Cantor diagonal procedure, we get a subsequence such that .
Then, it is standard to extend to a bounded linear operator on the whole space . Clearly,

4. Some Applications

It is well-known that weak convergences are ill behaved under nonlinear transformations (in contrast to their good behaviour under linear transformations). Only in some particular cases of compensation, it is even possible to pass to the limit in a product of two weakly converging sequences.

The prototype of this compensation effect is Tartar-Murat's div-curl lemma (cf. [24, Theorem  7.1]).

For simplicity, consider two vector-valued sequences, and , converging to zero weakly in , such that and are both contained in a compact set of (which then implies that they converge to zero strongly in ).

We can define , which (on a subsequence) defines a    -measure . By the localisation principle [2, Theorem  1.6] and [14, Theorem  2], as the above relations can be written in the form ( are constant matrices with all entries zero except and ) the corresponding -measure satisfies . After straightforward calculations this shows that weak in the sense of Radon measures (and therefore in the sense of distributions as well).

For the above, one has used only the nondiagonal blocks of corresponding to products of and ; in fact, the calculation shows that , which gives the above result.

In order to get a similar result using -distributions, we first show that the following localisation principle holds.

Theorem 4.1. Assume that in and in , for some , such that they satisfy
Take an arbitrary sequence bounded in , and by denote the -distribution corresponding to some subsequences of sequences and . Then, in the sense of distributions on , the function being the symbol of the linear partial differential operator with coefficients.

Proof. In order to prove the theorem, we need a particular multiplier, the so called (Marcel) Riesz potential , and the Riesz transforms [19, V.1,2]. We note that [id.,V.2.3] From here, using the density argument and the fact that is bounded from to itself, we conclude that .
We should prove that the -distribution corresponding to (the chosen subsequences of) and satisfies (4.4). To this end, take the following sequence of test functions: where and , . Then, apply the right-hand side of (4.3), which converges strongly to 0 in by the assumption, to a weakly converging sequence in the dual space .
We can do that since is a bounded sequence in .
Indeed, is bounded in any ( ). By the well-known fact [19, Theorem  V.1] that is bounded from , is bounded in for all sufficiently large . Then, take and due to the compact support of , one has that boundedness implies the same in . On the other hand, is bounded from to itself, for any , thus; is bounded in .
Therefore, one has (the sequence is bounded and 0 is the only accumulation point, so the whole sequence converges to 0)
Concerning the left-hand side of (4.3), according to (4.5), one has The first term on the right is of the form of the right-hand side of (2.2). The integrand in the second term is supported in a fixed compact and weakly converging to 0 in , so strongly in , where is such that (i.e., ). Of course, the argument giving the boundedness of in above applies also to instead of .
Therefore, from (4.7) and (4.8), one concludes (4.4).

Remark 4.2. Notice that the assumption of the strong convergence of in can be relaxed to local convergence, because in the proof we used a cutoff function .
Let us return to the simple example from the beginning of this section; consider two vector-valued sequences and , this time converging to zero weakly in and , respectively. Assume that the sequence is bounded in , and in (thus precompact in , and , resp.).
Then, the sequence is bounded in so also in (Radon measures) and by weak compactness it has a weakly converging subsequence. However, we can say more—the whole sequence converges to zero.
Denote by the -distribution corresponding to (some sub)sequences (of) and .
Since is bounded in , and is bounded in , they are weakly precompact, while the only possible limit is zero, so Now, from the compactness properties of the Riesz potential (see the proof of previous theorem), we conclude that, for every and , the following limit holds in : Multiplying (4.10) first by and then by , integrating over and passing to the limit, we conclude from (2.2), due to the arbitrariness of and :
Next, take
From (4.9), we get for . Rewriting it in the integral formulation, we obtain, from (2.2), From the algebraic relations (4.11) and (4.14), we can easily conclude implying that the distribution is supported on the set , which implies .
After inserting in the definition of -distribution (2.2), we immediately reach the conclusion. This proof is similar to the case, but it should be noted that there we had used only a nondiagonal block of    -measure, which corresponds to the only available    -distribution.
There is no reason to limit oneself to two dimensions; take and converging weakly to zero in and , and by denote matrix -distribution corresponding to some chosen subsequences of and .
From strongly in , for and , one has as in (4.10) that After forming a product with , integrating and passing to the limit, we conclude that namely, that the columns of are perpendicular to .
On the other hand, from strongly in , in an analogous way, we conclude that, for each row (denoted by ) of , for all , one has so the rows of are proportional to and (a rank-one matrix), being the constants of proportionality. So, the columns of are proportional to , while earlier we showed that they are perpendicular to . Thus, , which implies the convergence , as in the two-dimensional situation.

The above result is the well-known Murat’s div-curl lemma in the -setting [24, 25], which we state as a theorem.

Theorem 4.3. Let and be vector-valued sequences converging to zero weakly in and , respectively. Assume that the sequence is bounded in and the sequence is bounded in .
Then, the sequence converges to zero in the sense of distributions (or vaguely in the sense of Radon measures).

Acknowledgment

Originally, Theorem 2.1 was proved only in the case . The authors would like to thank Martin Lazar for pointing out the possibility to extend the theorem to more general values of . They wish to thank the referee for numerous suggestions which helped them to improve the final version of this paper. The work of N. Antonić is supported in part by the Croatian MZOS through projects 037-0372787-2795 and 037-1193086-3226. D. Mitrović is engaged as a part-time researcher at the University of Bergen, in the framework of project Mathematical Modelling over Multiple Scales of the Research Council of Norway. He gratefully acknowledges the support of the Research Council of Norway.

References

  1. L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, vol. 74, American Mathematical Society, 1990.
  2. L. Tartar, “H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 115, no. 3-4, pp. 193–230, 1990. View at Google Scholar · View at Zentralblatt MATH
  3. P. Gérard, “Microlocal defect measures,” Communications in Partial Differential Equations, vol. 16, no. 11, pp. 1761–1794, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. Aleksić, D. Mitrović, and S. Pilipović, “Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media,” Journal of Evolution Equations, vol. 9, no. 4, pp. 809–828, 2009. View at Publisher · View at Google Scholar
  5. J. Aleksić and D. Mitrović, “On the compactness for two dimensional scalar conservation law with discontinuous flux,” Communications in Mathematical Sciences, vol. 7, no. 4, pp. 963–971, 2009. View at Google Scholar · View at Zentralblatt MATH
  6. N. Antonić and M. Lazar, “Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 6, pp. 4500–4512, 2010. View at Publisher · View at Google Scholar
  7. N. Antonić and Marko Vrdoljak, “Parabolic H-convergence and small-amplitude homogenization,” Applicable Analysis, vol. 88, no. 10-11, pp. 1493–1508, 2009. View at Publisher · View at Google Scholar
  8. R. V. Kohn, “The relaxation of a double-well energy,” Continuum Mechanics and Thermodynamics, vol. 3, no. 3, pp. 193–236, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Lewandowski, “Vorticities in a LES model for 3D periodic turbulent flows,” Journal of Mathematical Fluid Mechanics, vol. 8, no. 3, pp. 398–422, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. Métivier and S. Schochet, “Trilinear resonant interactions of semilinear hyperbolic waves,” Duke Mathematical Journal, vol. 95, no. 2, pp. 241–304, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Mielke, “Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner-Husimi transforms,” Archive for Rational Mechanics and Analysis, vol. 181, no. 3, pp. 401–448, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. Mitrović and I. Ivec, “A generalization of H-measures and application on purely fractional scalar conservation laws,” Communications on Pure and Applied Analysis, vol. 10, p. 11, 2011. View at Google Scholar
  13. E. Y. Panov, “Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,” Archive for Rational Mechanics and Analysis, vol. 195, no. 2, pp. 643–673, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. N. Antonić, “H-measures applied to symmetric systems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 126, no. 6, pp. 1133–1155, 1996. View at Google Scholar · View at Zentralblatt MATH
  15. N. Antonić and M. Lazar, “A parabolic variant of H-measures,” Annali dell'Universitá di Ferrara, vol. 54, no. 2, pp. 183–201, 2008. View at Publisher · View at Google Scholar
  16. L. Tartar, The General Theory of Homogenization, vol. 7, Springer, Berlin, Germany, 2009. View at Publisher · View at Google Scholar
  17. G. O. Okikiolu, Aspects of the Theory of Bounded Integral Operators in Lp-Spaces, Academic Press, London, UK, 1971.
  18. L. Grafakos, Classical Fourier Analysis, vol. 249, Springer, New York, NY, USA, 2nd edition, 2008.
  19. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970.
  20. N. Antonić and M. Lazar, “H-measures and variants applied to parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 207–225, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. E. Y. Panov, “Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property,” Journal of Mathematical Sciences, vol. 159, no. 2, pp. 180–228, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S. A. Sazhenkov, “The genuinely nonlinear Graetz-Nusselt ultraparabolic equation,” Rossiĭskaya Akademiya Nauk, vol. 47, no. 2, pp. 431–454, 2006 (Russian), Translation in Siberian Mathematical Journal, vol. 47, no. 2. pp. 355–375, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. Hörmander, The Analysis of Linear Partial Differential Operators I–IV, Springer, 1985.
  24. F. Murat, “A survey on compensated compactness,” in Contributions to Modern Calculus of Variations, vol. 148, pp. 145–183, Longman Sci. Tech., Harlow, UK, 1987. View at Google Scholar
  25. F. Murat, “Compacité par compensation,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 5, no. 3, pp. 489–507, 1978. View at Google Scholar · View at Zentralblatt MATH