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Über dieses Buch

This book covers a diverse range of topics in Mathematical Physics, linear and nonlinear PDEs. Though the text reflects the classical theory, the main emphasis is on introducing readers to the latest developments based on the notions of weak solutions and Sobolev spaces.

In numerous problems, the student is asked to prove a given statement, e.g. to show the existence of a solution to a certain PDE. Usually there is no closed-formula answer available, which is why there is no answer section, although helpful hints are often provided.

This textbook offers a valuable asset for students and educators alike. As it adopts a perspective on PDEs that is neither too theoretical nor too practical, it represents the perfect companion to a broad spectrum of courses.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract
In this chapter, necessary tools from Mathematical Analysis are recalled. These include an introduction to differential forms, Stoke’s Theorem and its variants, convolutions, and mollifiers.
Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Chapter 2. Distributions, Sobolev Spaces and the Fourier Transform

Abstract
This chapter begins with the introduction and properties of the Fourier transform, and then distributions and weak derivatives are introduced. Second half of the chapter is devoted to Sobolev spaces and their properties.
Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Chapter 3. Common Methods

Abstract
This chapter introduces the necessary tools from functional analysis. We begin with the notion of weak (and weak) convergence–first in Hilbert, then in Banach spaces. These tools are necessary to introduce and study the separation of variables technique and the Galerkin method.
Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Chapter 4. Elliptic Equations

Abstract
This chapter is devoted to the study of elliptic equations. We begin with classical theory of harmonic functions, and then the modern approach to weak solutions (variational approach, Lax–Milgram’s Lemma) is introduced.
Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Chapter 5. Evolution Equations

Abstract
This chapter deals with evolution equations, i.e., equations depending on a time variable. It begins with first order equations and the theory of characteristics. Then, the wave equation is introduced, leading to a systematic study of hyperbolic equations (Fourier transform methods and energy estimates). Finally, parabolic equations and the regularity of their weak solutions are investigated.
Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

Backmatter

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