Liu X., Schulze B.-W. Boundary Value Problems with Global Projection Conditions

Springer, 2018. — 421 p. — (Operator Theory: Advances and Applications 265). — ISBN: 978-3-319-70113-4.This book presents boundary value problems for arbitrary elliptic pseudo-differential operators on a smooth compact manifold with boundary. In this regard, every operator admits global projection boundary conditions, giving rise to analogues of Toeplitz operators in subspaces of Sobolev spaces on the boundary associated with pseudo-differential projections. The book describes how these operator classes form algebras, and establishes the concept for Boutet de Monvel’s calculus, as well as for operators on manifolds with edges, including the case of operators without the transmission property. Further, it shows how the calculus contains parametrices of elliptic elements. Lastly, the book describes natural connections to ellipticity of Atiyah-Patodi-Singer type for Dirac and other geometric operators, in particular spectral boundary conditions with Calderón-Seeley projections and the characterization of Cauchy data spaces.Pseudo-differential operators
BVPs with the transmission property
Shapiro–Lopatinskii ellipticity
Toeplitz boundary value problems
Cutting and pasting of elliptic operators, Cauchy data spaces
The cone algebra
The edge algebra
Edge-ellipticity
Toeplitz edge problems
The edge approach to BVPs
Boundary ellipticity
Toeplitz boundary value problems without the transmission property
Examples, applications and remarks