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This book presents important contributions to modern theories concerning the distribution theory applied to convex analysis (convex functions, functions of lower semicontinuity, the subdifferential of a convex function). The authors prove several basic results in distribution theory and present ordinary differential equations and partial differential equations by providing generalized solutions. In addition, the book deals with Sobolev spaces, which presents aspects related to variation problems, such as the Stokes system, the elasticity system and the plate equation. The authors also include approximate formulations of variation problems, such as the Galerkin method or the finite element method.

The book is accessible to all scientists, and it is especially useful for those who use mathematics to solve engineering and physics problems. The authors have avoided concepts and results contained in other books in order to keep the book comprehensive. Furthermore, they do not present concrete simplified models and pay maximal attention to scientific rigor.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
We give a brief motivation for the concept of distributions. We explain why this concept is useful in applied mathematics. We present some landmarks in the history of this subject. Then we present the main topics and results in the theory of distributions.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 2. Preliminaries

Abstract
This chapter presents some concepts that are used in the next chapters. One section is dedicated to seminorms and locally convex spaces, as well as to convex, balanced and absorbing sets. Another section presents duals and reflexive spaces. Finally, the inductive limit topology is presented.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 3. Convex and Lower-Semicontinuous Functions

Abstract
In this chapter, one section is about convex functions and their properties. Another section is about lower-semicontinuous functions and their properties in compact topological spaces and in Banach spaces. The final section presents some properties of functions that are both convex and lower-semicontinuous. More precisely, some conditions are discussed that are needed for such a function to take a minimum value on a Banach space.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 4. The Subdifferential of a Convex Function

Abstract
In this chapter, the first section presents the definitions of Gateaux differentiable functions and of Frechet differentiable functions and the concept of subdifferentiability. Monotone and maximal monotone operators are defined. Minty’s theorem is proved. The subdifferential is shown to be a maximal monotone operator. The conjugate function is used to transform a minimization problem into a maximization problem and conversely. Finally, the additivity of the subdifferential is studied.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 5. Evolution Equations

Abstract
This chapter presents some results about evolution equations. Moreover, it presents the definition and the main properties of the resolvent, the Yosida approximation and the principal section of a maximal monotone operator. A stability result is shown for the solution of the Cauchy problem associated to an evolution equation based on the inequality of Gronwall.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 6. Distributions

Abstract
First of all, we introduce some fundamental spaces that are needed in the theory of distributions. Then we introduce the space of distributions. Then we discuss about the derivative of a distribution, the primitive of a distribution and higher-order primitives. Finally, we present the direct product of distributions, the convolution of distributions and their properties.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 7. Tempered Distributions

Abstract
First of all, we introduce the Schwartz space of infinitely differentiable functions that are rapidly decreasing at infinity. Then we define tempered distributions. We consider the Fourier transform and its properties. Finally, the Paley-Wiener-Schwartz theorem is discussed.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 8. Differential Equations in Distributions

Abstract
This chapter presents both ordinary and partial differential equations in distributions. Linear differential equations with constant coefficients are discussed in the framework of the theory of distributions. Hyperbolic, parabolic and elliptic partial differential equations are solved by means of the Fourier transform. The Cauchy problem is discussed.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 9. Sobolev Spaces

Abstract
We define some Sobolev spaces and study some of their properties, such as the inequality of Poincare, trace theorems or embedding results. Then we present the Littlewood-Paley decomposition and Besov spaces.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 10. Variational Problems

Abstract
We present the existence and uniqueness of the solutions of a large class of variational problems. We apply the Lax-Milgram theorem for the variational formulations of the Stokes system, the elasticity system and the plate equation. Then we discuss the approximation of variational problems by means of the Galerkin method and by means of the finite element method.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 11. On Some Spaces of Distributions

Abstract
We present the spaces \(\mathcal {D}_{L^p}\) and their properties. We also present their duals and their properties. Then we ask which is the most general space of distributions operating continuously on \(\mathcal {S}'\) by convolution and such that the Fourier transform maps convolution products into the product of the corresponding Fourier transforms. So we present the space \(\mathcal {O}'_C\) and its properties.
Adina Chirilă, Marin Marin, Andreas Öchsner

Chapter 12. On Some Differential Operators

Abstract
We present local and pseudolocal operators, give examples and study their properties. Then we discuss the hypoelliptic partial differential operators and their properties. Finally, we study the existence of fundamental solutions.
Adina Chirilă, Marin Marin, Andreas Öchsner

Backmatter

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