Leray J. Hyperbolic Differential Equations

Institute for Advanced Study, 1953. — 242 p.Linear hyperbolic equations with constant coefficients and symbolic calculus with several variablesThe symbolic calculus
Fourier and Laplace transforma
Definition and main properties of the symbolic calculus
Examples
Symbolic product by a function of +-p₁²+-...+-p_l²
Preliminary
Symbolic product by a function of p₁² +... + P_l²
Symbolic product by a function of -p₁² + p₂²+... + P_l²
Symbolic product by a^β(p), where a is a polynomial and β a complex number; the case β=-1
The real projection of the algebraic manifold a(ζ) = 0 and the complement Δ(a) of the closure of its projection
The director cone Γ(a) of Δ(a) and its dual C(a)
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
The elementary solution
Conclusions
Symbolic product by 1/a(p), when a(p) is a homogeneous polynomial
The exterior differential calculus
Herglotz's formula
The case: l even, m-l > l (Herglotz)
The case: l odd, m-l > l (Petrowsky)
The general case
Example: the waves equation
Linear hyperbolic equations with variable coefficientsThe existence of global solutions on a vector spaceThe matrices B defining norms for which a given matrix A is hermitian
The operators B defining norms for which the hermitian part of a given operator A is bounded
A priori bound for the local solutions of the hyperbolic equation
Existence theorems
The inverses of a hyperbolic operator on a vector spaceThe cones whose sheets separate the sheets of a given cone
The hyperbolic operators of order m-l whose product by a given hyperbolic operator of order m has a positive hermitian part
The inverses of a regularly hyperbolic operator
Emission and dependence domain
The inverses of a hyperbolic operator on a manifold
Hyperbolic operators and emission
The inverses of a hyperbolic operator
The elementary solutions
Cauchy's problem
Hyperbolic systems
Notation and results
The proof of the preceding statements
Non-linear equations systemsPreliminary: Quasi-linear equations and systems
Non-linear equations
Non-linear systems