In this essay we present some elementary measure theory and some theory of Lp-spaces with the goal of proving the Riesz-Thorin interpolation theorem. / I denna uppsats presenteras grundläggande måtteori och något kring teorin om Lp-rum med målet att bevisa Riesz-Thorins interpolationssats.

In this essay we present an introduction to real analysis, with the purpose of proving the Implicit Function Theorem. Our proof relies on other well-known theorems in set theory and real analysis as the Heine-Borel Covering Theorem and the Inverse Function Theorem. / I denna uppsats ger vi en introduktion till reel analys, med syftet att bevisa den implicita funktionssatsen. Vårt bevis bygger på andra välkända satser i mängdteori och reel analys som Heine-Borels övertäckningssats och inversa funktionssatsen.

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This Licentiate thesis is focussed on some new questions in homogenization theory, which have been motivated by some concrete problems in tribology. From the mathematical point of view, these questions are equipped with scales of Reynolds equations with rapidly oscillating coefficients. In particular, in this Licentiate thesis we derive the corresponding homogenized (averaged) equation. We consider the Reynolds equations in both the stationary and unstationary forms to analyze the effect of surface roughness on the hydrodynamic performance of bearings when a lubricant is flowing through it. In Chapter 1 we describe the possible types of surfaces a bearing can take. Out of these, we select two types and derive the appropriate Reynolds equations needed for their analysis. Chapter 2 is devoted to the derivation of the homogenized equations, associated with the stationary forms of the compressible and incompressible Reynolds equations. We derive these homogenized equations by using the multiple scales expansion technique. In Chapter 3 the homogenized equations for the unstationary forms of the Reynolds equations are considered and some numerical results based on the homogenized equations are presented. In chapter 4 we consider the equivalent minimization problem for the unstationary Reynolds equation and use it to derive a homogenized minimization problem. Finally, we obtain both the lower and upper bounds for the derived homogenized problem. / <p>Godkänd; 2007; 20070523 (ysko)</p>

This Licentiate Thesis is devoted above all to the investigation of variational inequalities. Chapter 1 deals with linear elliptic variational inequalities, where the operator is degenerated or singular, which involves the use of some weighted Sobolev spaces. It is shown in several examples how to interprete the (weak) solution of such variational inequality, if it is regular. In the next chapters, parabolic variational inequalities or equations on non-cylindrical domains are considered and the existence of a (weak) solution is proved by a generalization of the so-called method of Rothe. Chapter 2 is devoted to nonlinear parabolic inequalities with strongly elliptic part, while Chapter 3 deals with a linear parabolic equation, in which some singularities appear at du/dt as well as in the elliptic part, which involves the use of some weighted Sobolev spaces. In Chapter 4, the approach of Chapter 3 is extended from equations to linear singular parabolic inequalities. / <p>Godkänd; 2006; 20070110 (haneit)</p>

This Licentiate Thesis deals with parabolic problems on non-cylindrical domains. The existence and uniqueness of the corresponding initial- boundary value problem is proved by the method of Rothe, which - for the case of non-cylindrical domains - has to be appropriately generalized and applied. In Chapter 1 the Dirichlet problem for a linear operator of order 2k is investigated. Chapter 2 deals again with linear operators, but having some singularities at du/dt as well as in the elliptic part, which involves the use of some weighted Sobolev spaces. Chapter 3 is devoted to operators which are nonlinear in their elliptic part. In the last chapter, the so- called tranformation method, introduced in [3] and which allows to transform a parabolic problem on a non-cylindrical domain to a cylindrical one, is extended from strongly elliptic linear operators to operators, which are nonlinear and singular. / <p>Godkänd; 2006; 20070110 (haneit)</p>

This Licentiate Thesis consists of four chapters, which deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. Chapter 1 is an introduction, where, in particular, the importance to study function spaces with weights is discussed and motivated. In Chapter 2 we consider and analyze some results of L. D. Kudryavtsev, where he investigated one dimensional Sobolev spaces. Moreover, in this chapter we present and prove analogous results by B. L. Baidel'dinov for generalized Sobolev spaces. These results are crucially for the proofs of the main results of this Licentiate Thesis. In Chapter 3 we prove some embedding theorems for these new generalized Sobolev spaces. The main results of Kudryavtsev and Baidel'dinov about characterization of the behavior of functions at a singularity take place in weak degeneration of spaces. However, with the help of our new embedding theorems we can extend these results to the case of strong degeneration. In Chapter 4 we prove some new estimates for each function in a Tchebychev system. In order to be able to study also compactness of the embeddings from Chapter 3 such estimates are crucial. I plan to study this question in detail in my further PhD studies. / <p>Godkänd; 2007; 20071107 (ysko)</p>