Generalized functions in mathematical physics. Main ideas and concepts A.S. Demidov Contents Preface vii Notation xi Chapter 1. Introduction to problems of mathematical physics 1 1. Temperature at a point? No! In volumes contracting to the point 1 2. The notion of δ-sequence and δ-function 4 3. Some spaces of smooth functions. Partition of unity 6 4. Examples of δ-sequences 10 5. On the Laplace equation 11 6. On the heat equation 19 7. The Ostrogradsky–Gauss formula. The Green formulae and the Green function 28 8. The Lebesgue integral1) 31 p p 9. The spaces L and Lloc 40 1 ∞ 10. Functions of Lloc as linear functional on C0 45 11. Simplest hyperbolic equations. Generalized Sobolev solutions 46 Chapter 2. The spaces D[, D# and D0. Elements of the distribution theory (generalized function in the sense of L. Schwartz) 59 12. The space D[ of the Sobolev derivatives 59 13. The space D# of generalized functions 63 14. The problem of regularization 66 15. Generalized functions with a point support. The Borel theorem 68 16. The space D0 of generalized functions (distributions by L. Schwartz) 72 v vi CONTENTS Chapter 3. The spaces Hs. Pseudodifferential operators 81 17. The Fourier series and the Fourier transform. The spaces S and S0 81 18. The Fourier–Laplace transform. The Paley–Wiener theorem 94 19. Fundamental solutions. Convolution 99 20. On spaces Hs 102 21. On pseudodifferential operators (PDO) 106 22. On elliptic problems 111 Addendum. Addendum A new approach to the theory of generalized functions (Yu.V. Egorov) 129 Bibliography 137 Index 141 Preface Presently, the notion of function is not as finally crystallized and definitely established as it seemed at the end of the 19th cen- tury; one can say that at present this notion is still in evolution, and that the dispute concerning the vibrating string is still going on only, of course, in different scientific circumstances, involving other personalities and using other terms. Luzin N.N. (1935) [42] It is symbolic that in that same year of 1935, S.L. Sobolev, who was 26 years old that time, submitted to the editorial board of the journal “Matematicheskiy sbornik” his famous work [61] and pub- lished at the same time its brief version in “Doklady AN SSSR” [60]. This work laid foundations of a completely new outlook on the con- cept of function, unexpected even for N.N. Luzin — the concept of a generalized function (in the framework of the notion of distribu- tion introduced later). It is also symbolic that the work by Sobolev was devoted to the Cauchy problem for hyperbolic equations and, in particular, to the same vibrating string. In recent years Luzin’s assertion that the discussion concern- ing the notion of function is continuing was confirmed once again, and the stimulus for the development of this fundamental concept of mathematics is, as it was before, the equations of mathematical physics (see, in particular, Addition written by Yu.V. Egorov and [10, 11, 16, 17, 18, 32, 49, 67]). This special role of the equa- tions of mathematical physics (in other words, partial differential equations directly connected with natural phenomena) is explained by the fact that they express the mathematical essence of the funda- mental laws of the natural sciences and consequently are a source and stimulus for the development of fundamental mathematical concepts and theories. vii viii PREFACE The crucial role in appearance of the theory of generalized func- tions (in the sense the theory of distributions) was played by J. Ha- damard, K.O. Friedrichs, S. Bochner, and especially to L. Schwartz, who published, in 1944–1948, a series of remarkable papers concern- ing the theory of distributions, and in 1950–1951 a two-volume book [54], which immediately became classical. Being a masterpiece and oriented to a wide circle of specialists, this book attracted the at- tention of many people to the theory of distributions. The huge contribution to its development was made by such prominent math- ematicians as I.M. Gel’fand, L. H¨ormander and many others. As a result, the theory of distributions has changed all modern analysis and first of all the theory of partial differential equations. Therefore, the foundations of the theory of distributions became necessary for general education of physicists and mathematicians. As for students specializing in the equations of mathematical physics, they cannot even begin any serious work without knowing the foundations of the theory of distributions. Thus, it is not surprising that a number of excellent monographs and textbooks (see, for example, [12, 22, 23, 25, 31, 40, 44, 54, 57, 59, 62, 68, 69]) are devoted to the equations of mathematical physics and distributions. However, most of them are intended for rather well prepared readers. As for this small book, I hope that it will be clear even to undergraduates majoring in physics and math- ematics and will serve to them as starting point for a deeper study of the above-mentioned books and papers. In a nutshell the book gives an interconnected presentation of a some basic ideas, concepts, results of the theory of generalized functions (first of all, in the framework of the theory of distributions) and equations of mathematical physics. Chapter 1 acquaints the reader with some initial elements of the language of distributions in the context of the classical equations of mathematical physics (the Laplace equation, the heat equation, the string equation). Here some basic facts from the theory of the Lebesgue integral are presented, the Riesz spaces of integrable func- tions are introduced. In the section devoted to the heat equation, the student of mathematics can get familiar with the method of dimensionality and similarity, which is not usually included in the university program for mathematicians, but which is rather useful on the initial stage of study of the problems of mathematical physics. PREFACE ix Chapter 2 is devoted to the fundamentals of the theory of distri- butions due to L. Schwartz. Section 16 is the most important. The approach to some topics can also be interesting for the experts. Chapter 3 acquaints the reader with some modern tools and methods for the study of linear equations of mathematical physics. The basics of the theory of Sobolev spaces, the theory of pseudodif- ferential operators, the theory of elliptic problems (including some elementary results concerning the index of elliptic operators) as well as some other problems connected in some way with the Fourier transform (ordinary functions and distributions) are given here. Now I would like to say a few words concerning the style of the book. A part of the material is given according to the scheme: defini- tion — theorem — proof. This scheme is convenient for presenting results in clear and concentrated form. However, it seems reason- able to give a student the possibility not only to study a priori given definitions and proofs of theorems, but also to discover them while considering the problems involved. A series of sections serves this purpose. Moreover, a part of the material is given as exercises and problems. Thus, reading the book requires, in places, a certain ef- fort. However, the more difficult problems are supplied with hints or references. Problems are marked by the letter P (hint on Parking for the solution of small Problems). The importance of numerous notes is essentially connected with a playful remark by V.F. D’yachenko: “The most important facts should be written in notes, since only those are read”. The notes are typeset in a small font and located in the text immediately after the current paragraph. I am very grateful to Yu.V. Egorov, who kindly agreed to write Addendum to the book. I would also like to acknowledge my grati- tude to M.S. Agranovich, A.I. Komech, S.V. Konyagin, V.P. Palam- odov, M.A. Shubin, V.M. Tikhomirov, and M.I. Vishik for the useful discussions and critical remarks that helped improve the manuscript. I am also thankful to E.V. Pankratiev who translated this book and produced the CRC. While preparing this edition, some corrections were made and the detected misprints have been corrected. A.S. Demidov Notation N = {1, 2, 3,... }, Z = {0, ±1, ±2,... } and Z+ = {0, ±1, ±2,... } are the sets of natural numbers, integers and non-negative integers. n n−1 X ×√Y is the Cartesian product of the sets X and Y , X = X × X. i = −1 is the imaginary unit (“dotted i”). ◦ı = 2πi the imaginary 2π (“i with a circle”). Rn and Cn are n-dimensional Euclidean and complex spaces; R = R1, C = C1 3 z = x + iy, where x = <z ∈ R and y = =z ∈ R. x < y, x ≤ y, x > y, x ≥ y are the order relations on R. a 1 means the a sufficiently large. {x ∈ X | P } is the set of elements which belong to X and have a property P . ]a, b] = {x ∈ R | a < x ≤ b};[a, b], ]a, b[ and [a, b[ are defined similarly. ∞ {an} is the sequence {an}n=1 = {a1, a2, a3,... }. f : X 3 x 7→ f(x) ∈ Y is the mapping f : X → Y , putting into correspondence to an element x ∈ X the element f(x) ∈ Y . 1A is the characteristic function of the set A, i.e. 1A = 1 in A and 1 = 0 outside of A. A π arccot α = − arctan α. 2 x → a means that the numerical variable x converges (tends) to a. =⇒ means “it is necessary follows”. ⇐⇒ means “if and only if” (“iff”), i.e an equivalence. A b Ω means A is compactly embedded in Ω (see Definition 3.2).