Pseudo-Differential Operators, Singularities, Applications, Softcover reprint of the original 1st ed. 1997 Operator Theory: Advances and Applications Series, Vol. 93

This book grew out of lecture notes based on the DMV seminar "Pseudo- Differential Operators, Singularities, Applications" held by the authors in Reisenburg-Günzburg, 12–19 July 1992. The modern theory of elliptic boundary value problems in domains having conical or edge singularities on the boundary as well as the classical theory of elliptic boundary value problems and the original Kondratiev theory are presented. This material forms the foundation for the second part of the book which contains a new construction of pseudo-differential operators with symbols corresponding to the singularities of the boundary of different dimensions. This allows in particular to obtain complete asymptotic expansions of solutions near these singularities.

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.