“7 t <} Hyperbolic Differwe ntial Equations + Jean Leray i Table of Contents 7 First part - Linear hyperbolic equations with constant > c oefficients and symbolic ca leplas with several variables 3S De 2 3 Introduction pe 4 Bibliography ‘ Chap. I The symbolic calculus ° Fourier and Laplace transforms ) Pe 7 Definition and main properties of the symbolic: calculus Pe, 16 pe 2h §3. BXamplés Chap. II Symbolic product by a function of + pe + eee + 7 ° Pe 26 §1. Preliminary §2, Symbolic product by a function of Ps + coe + 7) - Dp. 27 83. Symbolic product by a function of “ps + PS + eee + PS Pe 35 Chap. III Symbolic product by al? (p), where a is a polynomial and ‘A a complex number; the case Ig =-l. §1. The real projection of the algebraic manifold a(& ) = 0 and the com plement A(a) of the closure of its projection pe 6 $2, The director cone r (a) of /\(a) and its dual C(a).- pe 56 §3. The convex domain A las /3y ’) such that the operator b(p)a(p) is bounded for p ¢ A fas (Ps b) where b is a polynoniial ‘p. 61 $h. The elementary solution pe 66 §5. Conclusions pe 690. *38688 03-2 Zup4 Ib. ~——s Ghapter IV Symbolic product by a (p), when-a(p) is a homogeneous polynomial $1. The exterior differential calculus §2. Herglotz's formula $36 The case: £ even, m= £>1 (Herglotz) She The case: @ odd, m-_£> 1 (Petrowsky) §5. The general case 86. Example: the waves equation Second Part ~ Linear hyperbolic equations with variable coefficients 10h Introduction Pe Bibliography 108 Pe Chap. V The existence of global solutions on a vector space Introduction to Chap, V 110 $l. The matrices B defining norms for which a given matrix A is hermitian Pe 13 §2. The operators B defining norms for which the hermitian part of a given operator A is bounded §3. A priori bound for the local solutions of the hyperbolic equation She Existence theorems Chap. VI The inverses of a hyperbolic operator on a vecgor space Introduction to Chap..VI , De §1. The cones whose sheets separate the sheets of a given cone De §2. The hyperbolic operators of order m=1 whose product by a given hyperbolic operator of order m has a positive hermitian part. Pe The inverses of a regularly hyperbolic operator Pe Emission and dependence domain Pe The emission front is a locus of trajectories De Chap. VII - The inverses of a hyperbolic operator on a manifold §1. Hyperbolic operators and emission Pe §2. The inverses of a hyperbolic operator Pe §3. The elementary solutions De She Cauchy's problem | De Chapter VIII - Hyperbolic systems §1. Notation and results Pe §2. The proof of the preceding statements Pe Third Part - Non-linear equations and systems Introduction - Pe Sle Preliminary: Quasi~linear equations and systems §26 Non-linear equations §3. a Non~linear systems Pe 2. First part Linear hyperbolic equations with constant coefficients and symbolic caloulus with several variables. sroduction Since Hadamard gave his Yale lectures in 1920 about the hyperbolic differential equations of second order, many important papers about the equations of any order. have been published by Herglotz, Schauder, Petrowsky, Bureau, M, Riesz, and Gérding, ‘They used various but interesting methods _ and their results are inportent but incomplete, The first part of our lectures is related to the equations with constant coefficients; it goes beyond Petrowsky's and Garding* 8 results, and it improves their methods. Hadamard and Bureau used the Green's formulas by means of a duality it transforms a boundary value problem into the problen of finding a particular solution with a given singularity; this transformed problemi s easy and this particular solution 4s handy and important when the given problem is very simple; but generally this method’ is q. difficult ones Herglotz, Petrowsky and Gaxding use another duality: that between the inde= pendent variables and the derivations, which gives rise to the Fourler and Leplace transformation, More precisely Herglotz and Petrowsky applied the Heaviside calculus, that is to say the Laplace transformation to one vari@= able. (as a matter of fact to the time) and the Fourier transformation to the others (as a matter of fact to the space); how shocking in a relativistic @ eee 3e worldJ It is not astonishing that Garding obtained more complete results by applying the Laplace transformation at once to ail the variables. But he did not express all the results this transformation gives; for instance! he uses the diréctor cones | of the convex domains LV , ‘without studying these imortant convex domains Q; he defines some operators by means of the Laplace transformation and the others by means of the Riesz's analytical prolongation, whereas it 1s convenient to define and to study these -operators all togdther by means of the Laplace transformation. We do not transform any boundary problem; but by the Laplace transform= ation Chapter I defines and studies ‘throughout the symbolic calevlus of several variables; this calculus enables us to solve the Cauchy's boundary value problem for differential equations (and would also enable one to solve equations containing both derivatives and finite differences): -see Chap. VII, -$h, R Noe 107"108"109 and Chap, VIII, now 1135 a pe gagiyehge Chapter II gives2 in par2 ticular a new2, ‘geheral -and coneise expression igi ef the inverse of 2. - 2, =~ eee? os and of its powers, ox, ox, 0x t S c Chapter III studies the inverse of any polynomial of 3, eee, = ® *y Chapter IV, assuming that this polynomial is homogeneous, achieves Herglote-Petrowsky's ealculation of its elemeritary solution, h. Bibliography 1. Differential equations [1] Fl. Bureau, Lt integration des équations linéaires aux derivées partielles du second ordre et du type hyperbolique normal (mémoires de la société royale des sciences de Liége, vol. 3, 1938, Pe 3-67). [2] Fl. Bureau, Essai sur l'intégration des equations linéaires aux derivees partielles (Mémoires publies par 1'Academie royale de Belgique, Classe des Sciences, vole 15, 1936, ps 1-115). a Les solutions élémentaires des ‘equations linéaires aux e o S dérivées partielles (ibids). e e e Sur llintegration des équations linéaires aux derivees . e e partielles (Academie royale de Belgique, Bulletin classe aE a arent des Scienees, vole 22, 1936, pe 156-17k). Te SEE (3] Fl, Bureau, Sur le probléme de Cauehy pour les equations linéaires a ayx dérivees partielles totalement hyperboliques & un R S nombre impair de variables independantes (Académie royale de Belgique, Bulletin classe des Sciencesy vol, 33, WhT, Pe 587-670). [lh] L, Garding, Linear hyperbolic partial differential equations with constant coefficients (Acta maths, vol. 85, 1951, pe 1-62). [5] 4 Hadamard, Lectures on Cauchy's problem in linear partial differential 0 equations, (Yale, 1921). e e Le probleme de Cauchy et les e7q uations aux.der, ivee,s partielles - e e epa linéaires hyperboliques (Hermann, 1932). t 5. (6) P, Huwnbert and Machlan (and Colombo), Formylaire pour le caleul syn-~ bolique y Le calcul symbolique et ses applications ala physique mathemgtique, Memorial des Scjentes mathematiquess Vole 100, 19413 vol. 105, 19h:7. (7) G, Herglotz,; Uber die Integration linearer , partielles Differential- gleichangen mit konstanten Koeffizienten (Bericht tiber die Verhandlungen der S8chsischen Akademie Wissenschaften zu Leipsig, Math-Phys, Klasse, vol, 78, 1926, py 93-126; Pe 2872318; vol, 80, 1928, ps 69-116). (8) I, Petrowsky, Uber das Cauchyschs Problem fiir Systeme von partielten Differentisigleichungen, Recueil mathy (Mat, Sbornik) 2, vol, Lh, 1937, py» 815-868, [9] I, ‘Petrowsky, On the diffusion of waves and the lacunas for hyperbdlic equations, id., vol, 59, 1945, pe 289-370. [10] M, Riesz, Ltintégrale de Riemann-Liouville et le probleme de Cauchy, Aota maths, vol, 81, 1919, pe 1-223, 2. Basic theories (xg) S, Boehner, Volesungen iber Fouriersche Integrale (Leipzig, 1932; Chelsea, 1918). 139} S, Bochner and Chandrasckharen, Fourier transforms (Anma%e of Mathy, Prineeton). (a3 S» Bochner andW . J. Martin, Several complex variables (Prineeton, 1948). (e3 T, Carleman, Integrale de Fourier (Uppsala, 19h). 66 {17} Ge Doetsch, Theorie und Andwendung der Laplace=Transformation, (Springer, 1937) Handbuch der Laplace~transformation (Birkh#user, 1950) (18) Hardy, Proc. London math. soce, vole 1h (1925); vob. 27 (1927) pe hO1 Thorin, Meddelanden fran Lunds Univ. math. semin., Thesis, 19,8 [19] S. Lefschetz, L'analysis situs et la géometrie algebrique, (Gauthier= Villars, 1923) [20] RF, Riesz, Acta Math., vol. 1, 1918, p. 71 {21} Le Schwartz, Théorie des distributions, vol. 1 et 2 (Hermann, 1951) summed up in the Annales Univ. Grenoble, vols 21, 195, Pe 57 7hs 23, 197, 1 72h Y. Fourés=Bruhat, Comptes rendus Acad. Seiences, vol. 286, 1953 (22] TMtchmarsh, Introduction to the theory of Fourier integrals (Oxford, 1937) [23] ° A, Weil, L'intégration dans les groupes topologiques et ses applications (Hermann, 1910) [2h] D. Ve Widder, The Laplace transform (Princeton) (25) N, Wiener, The Fourler integral and certain of its applications (Cam= bridge, 1933) [26] H, Villat, Lecons sur le Caleul sympolique (faites £ la Serbonne; & paraitre). | (» CHAPTER I THE SYMBOLIC CALCULUS:..° §1. Fourier and Laplece transforms (This whole §1 is classical: see Bochner's books [13] and {14).) 1. Fourier transforma. (See the bibliography: [13], [15], [21], [22], [23], [25].) Let X and = be two vector spaces of the same finite dimension £ over the field of real numbers. Supnose they are dual: there is given mat a bilinear function x of x « X and «€ 7 ‘bes ® y) . ") soweee 8 ae ae . . yy _ the values of whtch are real and which satisfies: if x . » * O for somé-“ ...7 “ : st x and all » , thenx= 0; ifx. 7 = 0 for some » and all x, then y) = 0g (See: Bourbaki, Algebre lineaire.) , fe The coordinates X19 sev Xp of x and ij ao 2. of 7) are real numbers and will be so chosen that x04 =X Dy tee thy De e Let f(x), g(x) be functions with complex values, defined on X; let A( » ), YP ( 7) ) be functions with complex valuesy defined on 73 £(x)eg(x) is the product of f(x) and g(x); f(x}ee(x) = / veel E(xey) (9) 89 + 8 £ 4g the cone volution of f(x) and g(x). We use the norms 1 HE) II g = ae oS 88) [Mery dx p 1° for qs lorq=2 2 (x) 1} oO” Supe | £(x)| . -f- “ . ” taMe : re. a . “2 Beye - * Be We then have a) pp Stay sitll + HEM 6 \ It is easy to prove the following:’ If Wfll,< +o and gil, < + oo, then fg ily < fil, Hell, < + COS and if ffl ,<<+* co and {1 gl/5 < + 0» then [tells < {£14 Hells < + Oe Thus, by use of the convolution, the f(x) such that [/f ll, < + © constitute a ring and the f(x) such that neu o< to constitute a vector space over this ring. If i]£(x) Il, < + 0, then its Fourier transform F [f(x)] is the continu- ous and bounded function s( » ), whiéh is given by the formula {exp. A s eo): (142) 1,2 F (£f (x)] == AC y ) = So/aeo/lft (x) expeol (=727 1 xXe ¢ ) dey eeeA Xy,) « It can be proved that, if |! fll, and i gil, < + 00, then F text] = Fle]. F [2], afterwards that f(x) tl, = [AC M2 if 8 J te]: ‘this enables one to define ¥ [f] whenever I] f He < + o- Plancherel proved the following regarding this extension: F( £] is an isometric linear mapping of the Hilbert space of all funce tions f(x) such that //f i p< + © onto the Hilbert space of all functions AC » ) such tthhaatt [14 yall + oo (thus it is a unitary mappings) This mappin is given by (1-2) whenever both Yf i, and i fil, < +0 3; its inverse is given by -1 (1.3) FS [A( » )] = £(x) = fooeS/A> () expe(2 TA x + 7% Dd 4 eas dip whenever both // é N4 and ue Ilo < + We .ta ,iM“

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