Hyperbolic Differential Equations

✍ Scribed by Jean Leray

• Tongue: English
• Leaves: 242
• Category: Library

✦ Table of Contents

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