Distribution Theory Applied to Differential Equations

Description

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. Contents 1 Introduction 1.1 Initial Remarks References 2 Preliminaries 2.1 Introduction 2.2 Test Functions and Regularization 2.3 Seminorms and Locally Convex Spaces 2.3.1 Locally Convex Spaces 2.3.2 Convex and Balanced Sets 2.3.3 Absorbing Sets 2.4 Duals 2.4.1 Reflexive Spaces 2.5 The Inductive Limit Topology References 3 Convex and Lower-Semicontinuous Functions 3.1 Introduction 3.2 Convex Functions 3.3 Lower Semicontinuous Functions 3.4 Convexity and Lower Semicontinuity References 4 The Subdifferential of a Convex Function 4.1 Introduction 4.2 The Conjugate Function 4.3 The Additivity of the Subdifferential References 5 Evolution Equations 5.1 Introduction 5.2 The Resolvent and the Yosida Approximation References 6 Distributions 6.1 Fundamental Spaces in the Theory of Distributions 6.1.1 On Some Properties of the Spaces Cm(Ω) and Cinfty(Ω) 6.2 The Space of Distributions 6.3 The Dual of Cinfty 6.4 The Derivative of a Distribution 6.5 Distributions as Generalized Functions 6.6 On Some Spaces of Distributions 6.7 The Primitive of a Distribution 6.7.1 Structure Theorems 6.8 Extras 6.8.1 Higher-Order Primitives 6.8.2 The Local Structure of Distributions 6.9 Convolutions 6.9.1 The Direct Product of Distributions 6.9.2 Convolution of Distributions 6.9.3 Convolution of Functions and Distributions: Regularization 6.9.4 Convolution Maps References 7 Tempered Distributions 7.1 The Schwartz Space of Infinitely Differentiable Functions … 7.2 Tempered Distributions 7.3 The Fourier Transform in mathcalS(mathbbRn) 7.3.1 The Inverse Fourier Transform 7.3.2 Properties of the Fourier Transform 7.4 Fourier Transform of Tempered Distributions 7.5 The Fourier Transform of a Distribution with Compact Support 7.6 The Product of a Distribution by a Cinfty Function 7.7 The Space of Multipliers of mathcalS'(mathbbRn) 7.8 Some Results on Convolutions with Tempered Distributions 7.9 The Paley-Wiener-Schwartz Theorem 7.10 A Result on the Fourier Transform of a Convolution of Two Distributions References 8 Differential Equations in Distributions 8.1 Ordinary Differential Equations 8.1.1 Linear Differential Equations with Constant Coefficients 8.1.2 An Application 8.2 Partial Differential Equations 8.2.1 The Direct Product 8.2.2 Hyperbolic Partial Differential Equations 8.2.3 Parabolic Partial Differential Equations 8.2.4 Elliptic Partial Differential Equations 8.2.5 The Cauchy Problem 8.2.6 An Application References 9 Sobolev Spaces 9.1 The Sobolev Space H1(Ω) 9.2 The Sobolev Space Hm(Ω) 9.3 The Sobolev Space Wk,p(Ω) 9.4 The Sobolev Spaces Hs(mathbbRn) 9.5 Besov Spaces 9.5.1 The Nonhomogeneous Littlewood-Paley Decomposition 9.5.2 Definition and Properties 9.5.3 The Homogeneous Littlewood-Paley Decomposition and the Homogeneous Besov Spaces References 10 Variational Problems 10.1 Introduction 10.1.1 The Stokes System 10.1.2 The Elasticity System 10.1.3 The Plate Equation 10.2 The Approximation of Variational Problems 10.2.1 The Galerkin Method 10.2.2 The Finite Element Method References 11 On Some Spaces of Distributions 11.1 The Spaces mathcalDLp 11.2 The Space mathcalO'C References 12 On Some Differential Operators 12.1 Local and Pseudolocal Operators 12.2 Hypoelliptic Partial Differential Operators 12.3 Existence of Fundamental Solutions References Index Index