Pseudo-Differential Operators, Singularities, Applications

Inhaltsverzeichnis

• 1 Sobolev spaces.
• 1.1 Fourier transform.
• 1.2 The first definition of the Sobolev space.
• 1.3 General definition of Sobolev spaces in ? n.
• 1.4 Representation of a linear functional over Hs.
• 1.5 Embedding theorems.
• 1.6 Sobolev spaces in a domain.
• 2 Pseudo-differential Operators.
• 2.1 The algebra of differential operators.
• 2.2 Basic properties of pseudo-differential operators.
• 2.3 Calculus of pseudo-differential operators.
• 2.4 Pseudo-differential operators on closed manifolds.
• 2.5 Gårding inequality.
• 3 Elliptic pseudo-differential operators.
• 3.1 Parametrices of the elliptic operators.
• 3.2 Elliptic operators on a manifold.
• 4 Elliptic boundary value problems.
• 4.1 Model elliptic boundary value problems.
• 4.2 Elliptic boundary value problems in a domain.
• 5 Kondratiev’s theory.
• 5.1 A model problem.
• 5.2 The general problem.
• 5.3 The boundary value problem in an infinite cone for operators with constant coefficients.
• 5.4 Equations with variable coefficients in an infinite cone.
• 5.5 The boundary value problem in a bounded domain.
• 6 Non-elliptic operators; propagation of singularities.
• 6.1 Canonical transformations and Fourier integral operators.
• 6.2 Wave fronts of distributions.
• 6.3 Wave fronts and Fourier integral operators.
• 6.4 Propagation of singularities.
• 6.5 The Cauchy problem for a strongly hyperbolic equation.
• 7 Pseudo-differential operators on manifolds with conical and edge singularities; motivation and technical preparations.
• 7.1 The general background.
• 7.2 Parameter-dependent pseudo-differential operators and operator-valued Mellin symbols.
• 8 Pseudo-differential operators on manifolds with conical singularities.
• 8.1 The cone algebra with asymptotics.
• 8.2 The algebra on the infinite cone.
• 9 Pseudo-differential operators on manifoldswith edges.
• 9.1 Pseudo-differential operators with operator-valued symbols.
• 9.2 The edge symbolic calculus.
• 9.3 Edge pseudo-differential operators.
• 9.4 Applications, examples and remarks.