Hyperbolic Differential Equations

First page First part - Linear hyperbolic equations with constant coefficients and symbolic calculus with several variables Introduction Bibliography Chap. I The symbolic calculus l. Fourier and Laplace transforma 2. Definition and main properties of the symbolic calculus 3. Examples Chap. II Symbolic product by a function of +-p12+-...+-p_l2 l. Preliminary 2. Symbolic product by a function of p12 + ... + P_l2 3. Symbolic product by a function of -p12 + p22+... + P_l2 Ch. III Symbolic product by a^β(p), where a is a polynomial and β a complex number; the case β=-1 l. The real projection of the algebraic manifold a(ζ) = 0 and the complement Δ(a) of the closure of its projection 2. The director cone Γ(a) of Δ(a) and its dual C(a) 3. The convex domain Δ_α(a,β,b) such that the operator b(p)a^β(p) is bounded for p in Δ_α(a,β,b) where b is a polynomial 4. The elementary solution 5. Conclusions Chap. IV Symbolic product by 1/a(p), when a(p) is a homogeneous polynomial 1. The exterior differential calculus 2. Herglotz's formula 3. The case: l even, m-l > l (Herglotz) 4. The case: l odd, m-l > l (Petrowsky) 5. The general case 6. Example: the waves equation Second Part - Linear hyperbolic equations with variable coefficients Introduction Bibliography Chap. V The existence of global solutions on a vector space Introduction to Chap. V 1. The matrices B defining norms for which a given matrix A is hermitian 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 4. Existence theorems Chap. VI The inverses of a hyperbolic operator on a vector space Introduction to Chap. VI 1. The cones whose sheets separate the sheets of a given cone 2. The hyperbolic operators of order m-l whose product by a given hyperbolic operator of order m has a positive hermitian part 3. The inverses of a regularly hyperbolic operator 4. Emission and dependence domain Chap. VII The inverses of a hyperbolic operator on a manifold 1. Hyperbolic operators and emission 2. The inverses of a hyperbolic operator 3. The elementary solutions 4. Cauchy's problem Chap. VIII Hyperbolic systems l. Notation and results 2. The proof of the preceding statements Third Part - Non-linear equations systems Introduction l. Preliminary: Quasi-linear equations and systems 2. Non-linear equations 3. Non-linear systems