Mathematical Analysis and Numerical Methods for Science and Technology (Volume 2: Functional and Variational Methods)

These 6 volumes - the result of a 10 year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences.

III. Functional Transformations.- A. Some Transformations Useful in Applications.- §1. Fourier Series and Dirichlet’s Problem.- 1. Fourier Series.- 1.1. Convergence in L2 (T).- 1.2. Pointwise Convergence on T.- 2. Distributions on T and Periodic Distributions.- 2.1. Comparison of D’(T) with the Distributions on ?.- 2.2. Principal Properties of D’(T).- 3. Fourier Series of Distributions.- 4. Fourier Series and Fourier Transforms.- 5. Convergence in the Sense of Césaro.- 6. Solution of Dirichlet’s Problem with the Help of Fourier Series.- 6.1. Dirichlet’s Problem in a Disk.- 6.2. Dirichlet’s Problem in a Rectangle.- §2. The Mellin Transform.- 1. Generalities.- 2. Definition of the Mellin Transform.- 3. Properties of the Mellin Transform.- 4. Inverse Mellin Transform.- 5. Applications of the Mellin Transform.- 6. Table of Some Mellin Transforms.- §3. The Hankel Transform.- 1. Generalities.- 2. Introduction to Bessel Functions.- 3. Definition of the Hankel Transform.- 4. The Inversion Formula.- 5. Properties of the Hankel Transform.- 6. Application of the Hankel Transform to Partial Differential Equations.- 6.1. Dirichlet’s Problem for Laplace’s Equation in?+3. The Case of Axial Symmetry.- 6.2. Boundary Value Problem for the Biharmonic Equation in ?+3, with Axial Symmetry.- 7. Table of Some Hankel Transforms.- Review of Chapter III A.- B. Discrete Fourier Transforms and Fast Fourier Transforms.- §1. Introduction.- §2. Acceleration of the Product of a Matrix by a Vector.- §3. The Fast Fourier Transform of Cooley and Tukey.- §4. The Fast Fourier Transform of Good-Winograd.- §5. Reduction of the Number of Multiplications.- 1. Relation Between the Discrete Fourier Transform and the Problem of Cyclic Convolution.- 2. Complexity of the Product of Two Polynomials.- 3. Application to the Cyclic Convolution of Order 2.- 4. Application to the Cyclic Convolution of Order 3.- 5. Application to the Cyclic Convolution of Order 6.- §6. Fast Fourier Transform in Two Dimensions.- §7. Some Applications of the Fast Fourier Transform.- 1. Solution of Boundary Value Problems.- 2. Regularisation and Smoothing of Functions.- 3. Practical Calculation of the Fourier Transform of a Signal.- 4. Determination of the Spectrum of Certain Finite Difference Operators and Fast Solvers for the Laplacian.- Review of Chapter III B.- IV. Sobolev Spaces.- §1. Spaces H1(?), Hm(?).- §2. The Space Hs(?n).- 1. Definition and First Properties.- 2. The Topological Dual of Hs(?n).- 3. The Equation (-? + k2)u = f in ?n , k ? ?\z0{.- §3. Sobolev’s Embedding Theorem.- §4. Density and Trace Theorems for the Spaces Hm(?), (m ? ? * = ?\z0{).- 1. A Density Theorem.- 2. A Trace Theorem for H1 (?+n).- 3. Traces of the Spaces Hm(?+n) and Hm(?).- 4. Properties of m-Extension.- §5. The Spaces H-m(?) for all m ? ?.- §6. Compactness.- §7. Some Inequalities in Sobolev Spaces.- 1. Poincaré’s Inequality for H01(?) (resp. H0m(?)).- 2. Poincaré’s Inequality for H1(?).- 3. Convexity Inequalities for Hm(?).- §8. Supplementary Remarks.- 1. Sobolev Spaces Wm, p(?).- 1.1. Definitions.- 1.2. Sobolev Injections.- 1.3. Trace Theorems for the Spaces Wm, p(?).- 2. Sobolev Spaces with Weights.- 2.1. Unbounded Open Sets.- 2.2. Polygonal Open Sets.- Review of Chapter IV.- Appendix: The Spaces Hs(?) with ? the “Regular” Boundary of an Open Set ? in ?n.- V. Linear Differential Operators.- §1. Generalities on Linear Differential Operators.- 1. Characterisation of Linear Differential Operators.- 2. Various Definitions.- 2.1. Leibniz’s Formula.- 2.2. Transpose of a Linear Differential Operator.- 2.3. Order of a Linear Differential Operator.- 3. Linear Differential Operator on a Manifold.- 4. Characteristics.- 4.1. Concept of Characteristics.- 4.2. Bicharacteristics.- 5. Operators with Analytic Coefficients. Theorems of Cauchy-Kowalewsky and of Holmgren.- §2. Linear Differential Operators with Constant Coefficients.- 1. Study of a l.d.o. with Constant Coefficients by the Fourier Transform.- 1.1. Existence of a Solution of Pu = f in the Space of Tempered Distributions.- 1.2. Example 1: The Laplacian.- 1.3. Elliptic and Strongly Elliptic Operators.- 1.4. Hypo-Elliptic and Semi-Elliptic Operators.- 1.5. Examples.- 1.6. Reduction of Operators of Order 2 in a Homogeneous, Isotropic “Medium”.- 2. Elementary Solutions of a l.d.o. with Constant Coefficients.- 2.1. Introduction.- 2.2. Elementary Solutions in S’ Examples.- 2.3. Elementary Solution with Support in a Salient Closed Convex Cone: Hyperbolic Operator.- 3. Characterisation of Hyperbolic Operators.- 3.1. Characteristics of a l.d.o. with Constant Coefficients.- 3.2. Algebraic Characterisation of Hyperbolic Operators.- 3.3. Hyperbolic Operators of Order 2.- 4. Parabolic Operators.- §3. Cauchy Problem for Differential Operators with Constant Coefficients.- 1. Cauchy Problem and the Elementary Solution in D’(?n× ?+).- 2. Propagation in Hyperbolic Cauchy Problems.- 3. Choice of a Functional Space: Well-Posed Cauchy Problem.- 4. Well-Posed Cauchy Problem in S’.- 5. Parabolic and Weakly Parabolic Operators.- 6. Study of the Particular Case P = ?/?t + P0.- 6.1. Analysis of One-Dimensional Case.- 6.2. Case in which P0 is Strongly Elliptic.- 6.3. Schrödinger Operator.- 7. Well-Posed Cauchy Problem in D’: Hyperbolic Operators.- §4. Local Regularity of Solutions*.- 1. Characterisation of Hypo-Ellipticity.- 1.1. Necessary Condition for Hypo-Ellipticity.- 1.2. Algebraic Transformation of the Necessary Condition for Hypo-Ellipticity.- 1.3. The Principal Result.- 2. Analyticity of Solutions.- 2.1. Statement of Results.- 2.2. Estimates of Analyticity.- 2.3. Generalisation: Gevrey Classes.- 3. Comparison of Operators.- 4. Local Regularity for Operators with Variable Coefficients and of Constant Force.- 5. Construction of an Elementary Solution.- §5. The Maximum Principle *.- 1. Prerequisites.- 2. Parabolic Maximum Principle and Dissipativity.- 3. Characterisation of Operators P Satisfying Maximum Principles.- 3.1. The Weak Maximum Principle.- 3.2. The Comparison Principle.- 3.3. The Strong Maximum Principle.- 3.4. The Principle of the Strong Parabolic Maximum.- Review of Chapter V.- VI. Operators in Banach Spaces and in Hilbert Spaces.- §1. Review of Functional Analysis: Banach and Hilbert Spaces.- 1. Locally Convex Topological Vector Spaces. Normed Spaces and Banach Spaces.- 2. Linear Operators.- 3. Duality.- 4. The Hahn-Banach Theorem and its Applications.- 4.1. Problems of Approximation.- 4.2. Problems of Existence.- 4.3. Problems of Separation of Convex Sets.- 5. Bidual, Reflexivity, Weak Convergence, Weak Compactness.- 5.1. Bidual.- 5.2. Reflexivity.- 5.3. Weak Convergence.- 5.4. Weak Compactness.- 5.5. Weak-Star Convergence.- 6. Hilbert Spaces.- 6.1. Definitions.- 6.2. Projection on a Closed Convex Set.- 6.3. Orthonormal Bases.- 6.4. The Riesz Representation Theorem. Reflexivity.- 7. Ideas About Functions of a Real or Complex Variable with Values in a Banach Space.- 7.1. Weak Topology.- 7.2. Weak Differentiability.- 7.3. Weak Holomorphy.- §2. Linear Operators in Banach Spaces.- 1. Generalities on Linear Operators.- 1.1. Domain, Kernel and Image of a Linear Operator.- 1.2. Nullity and Deficiency Indices.- 1.3. Basic Properties of Linear Operators.- 2. Spaces of Bounded Operators.- 2.1. Introduction.- 2.2. Various Concepts of Convergence of Operators.- 2.3. Composition and Inverse of Bounded Operators.- 2.4. Transpose of a Bounded Operator.- 2.5. Some Classes of Bounded Operators.- 2.6. Some Ideas on Functions of a Real or Complex Variable with Operator Values; Families of Operators.- 3. Closed Operators.- 3.1. Definition and Examples.- 3.2. Basic Properties.- 3.3. The Set ?(X, Y) of Closed Operators from X into Y.- 3.4. Transpose of a Closed Operator.- 3.5. Operators with Closed Image.- §3. Linear Operators in Hilbert Spaces.- 1. Bounded Operators in Hilbert Spaces.- 1.1. Adjoint Sesquilinear Form.- 1.2. Hermitian Operators.- 1.3. Orthogonal Projectors.- 1.4. Isometries and Unitary Operators.- 1.5. Hilbert-Schmidt Operators.- 2. Unbounded Operators in Hilbert Spaces.- 2.1. Adjoint of an Unbounded Operator.- 2.2. Symmetric Operators.- 2.3. The Cayley Transform.- 2.4. Normal Operators.- 2.5. Sesquilinear Forms and Unbounded Operators.- Review of Chapter VI.- VII. Linear Variational Problems. Regularity.- §1. Elliptic Variational Theory.- 1. The Lax-Milgram Theorem.- 2. First Examples.- 2.1. Example 1. Dirichlet Problem.- 2.2. Example 2. Neumann Problem.- 3. Extensions in the Case in which V and H are Spaces of Distributions or of Functions.- 4. Sesquilinear Forms Associated with Elliptic Operators of Order Two.- 5. Sesquilinear Forms Associated with Elliptic Operators of Order 2m.- 6. Miscellaneous Remarks.- 7. Application to the Solution of General Elliptic Problems (of Dirichlet Type).- §2. Examples of Second Order Elliptic Problems.- 1. Generalities.- 2. Examples of Variational Problems.- 2.1. Mixed Problem.- 2.2. Non-Local Boundary Conditions.- 3. Problems Relative to Integro-Differential Forms on ? + ?.- 3.1. Problem of the Oblique Derivative.- 3.2. Robin’s Problem.- 4. Transmission Problem.- 5. Miscellaneous Remarks.- 6. Application: Stationary Multigroup Equation for the Diffusion of Neutrons.- 7. Application: Statical Problems of Elasticity.- 7.1. Introduction.- 7.2. Variational Formulation.- 7.3. Korn’s Inequality.- 7.4. Application to Problem (2.39).- 7.5. Inhomogeneous Problem.- 8. Statical Problems of the Flexure of Plates.- §3. Regularity of the Solutions of Variational Problems.- 1. Introduction.- 2. Interior Regularity.- 3. Global Regularity of the Solutions of Dirichlet and Neumann Problems for Elliptic Operators of Order 2.- 4. Miscellaneous Results on Global Regularity.- 5. Green’s Functions.- 5.1. Case of the Laplacian in a Bounded Open Set ? with Dirichlet Condition.- 5.2. Some Other Particular Examples.- 5.3. Green’s Functions in a More General Setting.- Review of Chapter VII.- Appendix. “Distributions”.- §1. Definition and Basic Properties of Distributions.- 1. The Space D(?).- 1.1. Definition.- 1.2. Elementary Properties of the Convolution Product of Two Functions.- 1.3. A Procedure for the Construction of Functions of D(?).- 1.4. The Notion of Convergence in D(?).- 1.5. Some Inclusion and Density Properties.- 2. The Space D’(?) of Distributions on ?.- 2.1. Definition of Distributions and the Concept of Convergence in D’(?).- 2.2. First Examples of Distributions: Measures on ?.- 2.3. Differentiation of Distributions. Examples.- 2.4. Support of a Distribution. Distributions with Compact Support.- 3. Some Elementary Operations on Distributions.- 3.1. Product by a Function of Class ??.- 3.2. Primitives of a Distribution on a Interval of ?.- 3.3. Tensor Product of Two Distributions.- 3.4. Direct Image and Inverse Image of a Function and of a Distribution by a Function of Class ??.- 4. Some Examples.- 4.1. Primitives of the Dirac Measure.- 4.2. A Division Problem (Case n = 1).- 4.3. Derivative of a Function of ?n Discontinuous on a Surface.- 4.4. Distributions Defined by Inverse Image from Distributions on the Real Line.- §2. Convolution of Distributions.- 1. Convolution of a Distribution on ?n and a Function of D(?n).- 2. Convolution of Two Distributions of Which One (at Least) is with Compact Support.- 3. Distributions with Convolutive Supports.- 4. Convolution Algebras.- §3. Fourier Transforms.- 1. Fourier Transform of L1-Functions.- 2. The Space L(?n).- 3. Fourier Transform in L2.- 4. Fourier Transforms of Tempered Distributions.- 5. Fourier Transform of Distributions with Compact Support.- 6. Examples of the Calculation of Fourier Transforms.- 7. Partial Fourier Transform.- 8. Fourier Transform and Automorphisms of ?n: Homogeneous Distributions.- 8.1. Fourier Transform and Automorphisms of ?n.- 8.2. Homogeneous Distributions.- 9. Fourier Transform and Convolution. Spaces OM(?n) and O’C(?n).- 9.1. The Space OM(?n) ( = OM).- 9.2. The Space O’C.- 10. Fourier Transform of Tempered Measures.- 11. Distribution of Positive Type. Bochner’s Theorem.- 11.1. Functions of Positive Type.- 11.2. Distributions of Positive Type.- 12. Schwartz’s Theorem of Kernels.- 13. Some Distributions and Their Fourier Transforms.- Table of Notations.- of Volumes 1, 3–6.