An Introduction to Sobolev Spaces

The main objective of this chapter is to remind the reader of some basic notion and fundamental facts about real analysis and functional analysis required for the comprehension of the following chapters. It is assumed that readers are familiar with the concept of metric and normed spaces. The definitions of a metric space, convergence of a sequence in a metric space, completeness, compactness and the Heine Borel theorem are introduced in the first part. Here there are some well–known properties of topological concepts needed to recall. Next the notion of norm and normed spaces, equivalent norms, compactness and relatively compactness, Banach space, dual space, weak and weak* convergence are presented. Before the Hilbert spaces, inner products and inner product spaces are briefly expressed. Here the relation between the Banach and Hilbert spaces is given by some examples. Most of the theorems of that part are stated without proof since they can easily be found from any real or functional analysis book given in the reference part. Following the Fatou lemma and the Lebesgue dominated convergence theorem, the chapter ends with some important theorems based on fixed point properties on Banach spaces. For instance the Tychonoff fixed point theorem is an extension of the Schauder’s fixed point theorem and the Schauder’s theorem is an extension of the Brouwer fixed point theorem. Since the proofs of these theorems require additional knowledge, we refer the reader to the book by Papageorgiou and Winkert [1] and reference therein. In addition [2–7] may also be functional.

In this chapter, we deal with the spaces of Lp(Ω ). Here p may be a positive finite number or equal to ∞ and Ω is a measurable set. We start by introducing Lp(Ω ) space with its norm and provide some simple examples for different choices of Ω. Next part is devoted to several important inequalities which are stated in order as the Young, Holder and Minkowski inequalities with their corresponding reverse inequalities, the interpolation, Gronwall, Komornik and Nakao inequalities. Moreover, two types of the Green’s identities are presented. After investigating particular cases of finite p, we then consider the infinite case p = ∞, namely L∞ space. The Riesz-Fischer theorem is then analyzed with its proof in detail. In the remaining part, embedding property of Lp(Ω) space, L1loc(Ω) space, the space of continuous functions, C0∞ (Ω) space, Holder space and Cm([0; T];X) space with further theorems and properties are also given out in this chapter. Most of the theorems and propositions in that chapter are expressed with proofs. The statements without proof are addressed to the references given at the end of the book. Particularly [16–20] may be worthwhile.

This chapter is dedicated to applications of the weak derivatives. We extend the classical knowledge of derivative and provide basic properties of weak derivatives in this part. The concept of the weak derivative can be considered as a generalization of the classical derivative. More precisely there are functions which are not differentiable in the classical sense but weakly differentiable. Further properties and analysis are given in the examples of this section. For the applications of weak derivatives and further details, we refer the reader to [25–28].

Sobolev spaces were defined by the Russian mathematician Sergei Lvovich Sobolev (1908-1989) in the 1930s. Denoted by Wm;p (Ω) ; Sobolev spaces are the space of functions whose all m-th order generalized derivatives are in Lp (Ω) space and partial derivatives of these spaces satisfy certain integrability conditions. Notice that generalized derivative refers to the weak derivative which is defined in the previous chapter. In this part we present fundamental properties of Sobolev spaces with several examples. For further reading on Sobolev spaces we cite [29–32].

We start this chapter by introducing the embedded property of normed spaces. Next the Sobolev-Gagliardo-Nirenberg inequality and its proof are given. We then define the cone, Lipschitz and Cm class conditions and relation between them is analyzed by an example. The remaining part is dedicated to Sobolev embedding theorems with further inequalities such as Sobolev Poincar´e theorem, Poincar´e inequality and logarithmic Sobolev inequalities. Compact embedding and embedding in Hs(Rn) Sobolev spaces are investigated in the final part of the chapter.

In this chapter, we present the variable exponent Lebesgue spaces defined by Orlicz1) in 1931. Although the variable exponent Lebesgue spaces are introduced in 1931, it began to be actively studied in 1990s. We especially used the resources by Diening, Harjulehto, Hasto, Ruzicka 2011; Fan, Zhao 2001; Kovacik, Rakosnik 1991; Cruz-Uribe, Fiorenza 2013; Radulescu and Repovs 2015 for this section. The notion variable exponent Lebesgue and Sobolev spaces is directly related to the classical Lebesgue and Sobolev spaces where the constant p is replaced with the function p(.) which may depend on a variable. Further properties of these spaces are introduced and analyzed in that chapter.

In this chapter we analyze the solution spaces of some differential equations. Mainly the local existence of solutions of nonlinear Timoshenko equation and the existence of solutions of sixth order Boussinesq equation are analyzed. The main results of this part are based on the papers [34, 35]. For further details we also cite [36, 37] and the reference therein.