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2011 | Buch

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Lebesgue and Sobolev Spaces with Variable Exponents

verfasst von: Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Ruzicka

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The field of variable exponent function spaces has witnessed an explosive growth in recent years. For instance, a search for “variable exponent” in Mathematical Reviews yields 15 articles before 2000, 31 articles between 2000 and 2004, and 242 articles between 2005 and 2010. This is a crude measure with some misclassifications, but it is nevertheless quite telling.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Lebesgue spaces

Chapter 2. A Framework for Function Spaces

Abstract

In this chapter we study modular spaces and Musielak–Orlicz spaces which provide the framework for a variety of different function spaces, including classical (weighted) Lebesgue and Orlicz spaces and variable exponent Lebesgue spaces. Although our aim mainly is to study the latter, it is important to see the connections between all of these spaces. Many of the results in this chapter can be found in a similar form in [295], but we include them to make this exposition self-contained. Research in the field of Musielak–Orlicz functions is still active and we refer to [67] for newer results and references.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 3. Variable Exponent Lebesgue Spaces

Abstract

In this chapter we define Lebesgue spaces with variable exponents, \(L^{p(.)}\). They differ from classical \(L^p\) spaces in that the exponent p is not constant but a function from Ω to \([1,\infty]\). The spaces \(L^{p(.)}\) fit into the framework of Musielak–Orlicz spaces and are therefore also semimodular spaces.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 4. The Maximal Operator

Abstract

In the previous chapters we studied the spaces \(L^{p(.)}\) with general variable exponent p. We have seen that many results hold for fairly wild exponents, including discontinuous ones, in this general setting. We studied complete ness, separability, reflexivity, and uniform convexity.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 5. The Generalized Muckenhoupt Condition*

Abstract

The boundedness of the maximal operator M is closely linked to very impor tant properties of the spaces \(L^{p(.)}\). Indeed, we will see in Chaps. 6 and 8 that the boundedness ofM is needed for the Sobolev embeddings\(W^{1,p^{(.)}}\hookrightarrow L^{p^{*(.)}}\) , boundedness singular integrals on \(L^{p(.)}\) and Korn’s inequality.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 6. Classical Operators

Abstract

In this section we treat some of the most important operators of harmonic analysis in a variable exponent context. The results build on the boundedness of the maximal operator. We treat the Riesz potential operator, the sharp maximal function and singular integral operators in the three sections of the chapter. Several further operators are considered in Sect. 7.2. These results are applied in the second part of the book for instance to prove Sobolev embeddings and in the third part to prove existence and regularity of solutions to certain PDEs.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 7. Transfer Techniques

Abstract

This chapter is a collection of various techniques with the common theme “transfer”. In other words we study methods which allow us to take results from one setting and obtain corresponding results in another setting “for free”. The best known example of such a technique is interpolation, which has played an important unifying role in the development of the theory of constant exponent spaces [362–364].

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Sobolev spaces

Chapter 8. Introduction to Sobolev Spaces

Abstract

In this chapter we begin our study of Sobolev functions. The Sobolev space is a vector space of functions with weak derivatives. One motivation of studying these spaces is that solutions of partial differential equations belong naturally to Sobolev spaces (cf. Part III). In Sect. 8.1 we study functional analysis-type properties of Sobolev spaces, in particular we show that the Sobolev space is a Banach space and study its basic properties as reflexivity, separability and uniform convexity.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 9. Density of Regular Functions

Abstract

This chapter deals with the delicate question of when every function in a Sobolev space can be approximated by a more regular function, such as a smooth or Lipschitz continuous function. For the Lebesgue space, this question was solved in Theorem 3.4.12. An important fact is that log-Hölder continuity is sufficient for density of smooth functions. This is shown in Sect. 9.1. However, for the density question log-Hölder continuity is by no means necessary. Despite the contributions of many researchers, there remain substantial gaps in our understanding of this question. Indeed, it is fair to say that the results are in a transitory state and will hopefully be improved and unified in the future.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 10. Capacities

Abstract

Capacities are needed to understand point-wise behavior of Sobolev functions. They also play an important role in studies of solutions of partial differential equations. In this chapter we study two kinds of capacities: Sobolev capacity in Sect. 10.1 and relative capacity in Sect. 10.2. Both capacities have their advantages. The Sobolev capacity is independent of the underlying set, but extremal functions are difficult to find.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 11. Fine Properties of Sobolev Functions

Abstract

In this chapter we study fine properties of Sobolev functions. By definition, Sobolev functions are defined only up to Lebesgue measure zero and thus it is not always clear how to use their point-wise properties. We pick a good representative from every equivalence class of Sobolev functions and show that this representative, called quasicontinuous, has many good properties. Our main tools are the capacities studied in Chap. 10. Our results general- ize classical ones to the variable exponent case. In Sect. 11.1 we show that each Sobolev function has a quasicontinuous representatives under natural conditions on the exponent p and define a capacity based on quasicontinuous functions.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 12. Other Spaces of Differentiable Functions

Abstract

We have considered spaces of measurable functions in the first part of the book and spaces of functions with a certain number of derivatives in the second part. In the final chapter of this part we look at more general spaces with other kinds of differentiability.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Applications to partial differential equations

Chapter 13. Dirichlet Energy Integral and Laplace Equation

Abstract

For a constant \( q \ \epsilon \ (1, \infty)\), the Dirichlet energy integral is \( \int\limits_{\Omega}|\nabla u (x)|^q dx \). The problem is to find a minimizer for the energy integral among all Sobolev functions with a given boundary value function. The Euler–Lagrange equation of this problem is the q-Laplace equation,\(div(\mid\bigtriangledown u \mid^{q-2}\bigtriangledown u )\, = \,0\), which has to be understand in the weak sense.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Chapter 14. PDEs and Fluid Dynamics

Abstract

We use the theory of Calderón–Zygmund operators to prove regularity results for the Poisson problem and the Stokes problem, to show the solvability of the divergence equation, and to prove Korn’s inequality. These problems belong to the most classical problems treated in the theory of partial differential equations and fluid dynamics. It turns out that the treatment, especially of the whole space problems requires the notion of homogeneous Sobolev spaces, which have been studied in Sect. 12.2. The Poisson problem and the Stokes system are studied in the first two sections. After that we study the divergence equation and its consequences.

Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Růžička

Backmatter

Titel: Lebesgue and Sobolev Spaces with Variable Exponents
verfasst von: Lars Diening
Petteri Harjulehto
Peter Hästö
Michael Ruzicka
Copyright-Jahr: 2011
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-18363-8
Print ISBN: 978-3-642-18362-1
DOI: https://doi.org/10.1007/978-3-642-18363-8

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