Background

Variable exponent Lebesgue spaces appeared in the literature for the first time already in a 1931 article by W. Orlicz [11]. In this article the following question is considered: Let (pk) (with pk 1) and (xk) be sequences of real numbers such that

converges. What are the necessary and sufficient conditions on (yk) for to converge? It turns out that the answer is that should converge for some &lambda &gt 0 and pk =pk/(pk-1). This is exactly the Hölder inequality in the space

as we will see below. Orlicz also considered the variable exponent function space on the real line, and proved the Hölder inequality in this setting, too. However, after this one paper, Orlicz abandoned the study of variable exponent spaces, to concentrate on the theory of the function spaces that now bear his name. In the theory of Orlicz spaces, we define the space

as those measurable functions for which

for some &lambda &gt 0 (&phi has to satisfy certain conditions, which we will not get into here). If we allow &phi to depend also on x, we end up with a more general class of spaces. Again, assuming that &Phi satisfies certain conditions, such spaces are called modular. They were first systematically studied by H. Nakano [13,14]. In the appendix of the first of these books, Nakano mentions explicitly variable exponent Lebesgue spaces as an example of the more general spaces he considers [13, p. 284]. The dual property mentioned above is again observed.

Following the work of Nakano, modular spaces were investigated by several people, most importantly by groups at Sapporo, Japan, Voronez in the U.S.S.R. and Leiden, The Netherlands. Despite this broad interest, these spaces have not reached the same main-stream position as Orlicz spaces. Somewhat later, a more explicit version of such spaces, namely modular function spaces, were investigated by Polish matematicians, notably H. Hudzik [1]-[9] and J. Musielak [12]. For a nice presentation of the modular function spaces, see the book of J. Musielak [12]. This book, although not dealing specifically with the spaces that interest us, is still specific enough to contain several interesting results regarding variable exponent spaces. For instance we mention the uniform convexity of the generalized Orlicz space, see XX.

Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers, notably I. Sharapudinov. These investigations originated in a paper by I. Tsenov from 1961 [16]. The question raised by Tsenov and solved by Sharapudinov [15] is the minimization of

where u is a fixed function and v varies over a finite dimensional subspace of Lp(x)([a,b]). In the paper [15] Sharapudinov also introduces the Luxemburg norm for the Lebesgue space and shows that this space is reflexive if the exponent satisfies p- < 1 and p+<

In the mid-80's V. Zhikov started a new line of investigation, that was to become intimately related to the study of variable exponent spaces, namely he considered variational integrals with non-standard growth conditions.

The next major step in the investigation of variable exponent spaces was the comprehensive paper by O. Kovácik and J. Rákosník in the early 90's [10]. This paper established many of the basic properties of Lebesgue and Sobolev spaces. After this, this field of study lay dormant for nearly ten years. At the turn of the millennium several factors conjoined to start a period of systematic intense study of variable exponent spaces:

    • the "correct" condition for regularly varying exponents was found (the log-Hölder continuity condition), which allowed researchers to prove a multitude of results, starting with the boundedness of the maximal operator.

    • A connection was made with researcher investigating variational integrals with non-standard growth and coercivity conditions. At the same time it was found that these equations are related to modeling of so-called electrorheological fluids. Moreover, progress in physics over the past ten year have made the study of fluid mechanical properties of these fluids an important issue.

    • Several new groups emerged alongside the investigators of the Prague circle. These groups are connected to Freiburg, Helsinki Hiroshima, Lanzhaou, Parma and Tbliisi.

Many questions are now well understood, like the Hardy-Littlewood maximal operator,the Sobolev imbeddings and the density of smooth functions. It seems to be that the following condition for the exponent p is the borderline case:

This condition gives us many crucial classical tools, and without it for example the Hardy-Littlewood maximal operator does not need to be bounded, and smooth functions need not be dense in the Sobolev space.

Physical motivation for variable exponent Lebesgue and Sobolev spaces

Most materials can be modeled with sufficient accuracy using classical Lebesgue and Sobolev spaces, Lp and W1,p, where p is a fixed constant. For some materials with inhomogeneities, for instance electrorheological fluids, this is not adequate, but rather the exponent p should be able to vary. This leads us to the study of variable exponent Lebesgue and Sobolev spaces, Lp(x) and W1,p(x), where p is a real valued function.

Materials requiring such more advanced theory have been studied experimentally since the middle of last century. The first major discovery on electrorheological fluids is due to Willis Winslow in the year 1949. He noticed that such fluids (for instance lithiumpolymetachrylate) viscosity in an electrical field is inversely proportional to the strength of the field. The field induces string-like formations in the fluid, which are paralell to the field. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect.

Electrorheological fluids have been used in e.g. robotics and space technology. The experimental research has been done mainly in the USA, for instance in NASA laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids see [17, 18]

References

    1. H. Hudzik: On generalized Orlicz-Sobolev space, Funct. Approximatio Comment. Math. 4 (1976), 37-51.

    2. H. Hudzik: A generalization of Sobolev spaces. I, Funct. Approximatio Comment. Math. 2 (1976), 67-73.

    3. H. Hudzik: A generalization of Sobolev spaces. II, Funct. Approximatio Comment. Math. 3 (1976), 77-85.

    4. H. Hudzik: On problem of density of C∞0(&Omega) in generalized Orlicz-Sobolev space WkM(&Omega) for every open set &Omega &sub Rn, Comment. Math. Parce Mat. 20 (1977), 65-78.

    5. H. Hudzik: On continuity of the imbedding operation from WkM1(&Omega) into WkM2(&Omega), Funct. Approximatio Comment. Math. 6 (1978), 111-118.

    6. H. Hudzik: Density of C∞0(Rn) in generalized Orlicz-Sobolev space WkM2(Rn), Funct. Approximatio Comment. Math. 7 (1979), 15-21.

    7. H. Hudzik: The problem of separability, duality, reflexivity and comparison for generalized Orlicz-Sobolev space WkM(&Omega), Comment. Math. Parce Mat. 21 (1979), 315-324.

    8. H. Hudzik: Strict convexity of Musielak-Orlicz spaces with Luxemburg norm, Bull. Polish Acad. Sci. Math. 39 (1981), 235-247.

    9. H. Hudzik: Uniform convexity of Musielak-Orlicz spaces with Luxemburg norm, Comment. Math. Parce Mat. 23 ???

    10. O. Kovácik and J. Rákosník: On spaces Lp(x) and W1,p(x), Czechoslovak Math. J. 41(116) (1991), 592-618.

    11. W. Orlicz: Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200-211.

    12. J. Musielak: Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, 1983.

    13. H. Nakano: Modulared Semi-ordered Linear Spaces, Maruzen Co., Ltd., Tokyo, 1950.

    14. H. Nakano: Topology and Topological Linear Spaces, Maruzen Co., Ltd., Tokyo, 1951.

    15. I. I. Sharapudinov: On the topology of the space Lp(t)([0;1]), Matem. Zametki 26 (1978), no. 4, 613-632.

    16. I. V. Tsenov: Generalization of the problem of best approximation of a function in the space Ls, Uch. Zap. Dagestan Gos. Univ. 7 (1961), 25-37.

    17. T. C. Halsey: Electrorheological fluids, Science 258 (1992), 761-766.

    18. M. Ruzicka: Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.