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