Boundary value problems with global projection conditions
The present book is devoted to developing general concepts of ellipticity of boundary value problems (BVPs) and provides a self-contained resource for use by professional researchers. It consists of the three parts containing the following thirteen chapters: (1) Pseudo-differential operators, (2) BVPs with the transmission property, (3) Shapiro-Lopatinskii ellipticity, (4) Toeplitz boundary value problems, (5) Cutting and pasting of elliptic operators, Cauchy data spaces, (6) The cone algebra, (7) The edge algebra, (8) Edge-ellipticity, (9) Toeplitz edge problems, (10) The edge approach to BVPs, (11) Boundary ellipticity, (12) Toeplitz boundary value problems without the transmission property, (13) Examples, applications and remarks. The book is written in an accessible language and covers the fundamentals of the cone algebra and the edge algebra for elliptic pseudo-differential operators. In the first part of the book, the necessary tools on pseudo-differential operators are introduced. Then ellipticity on an open smooth manifold is defined, and parametrices within the algebra of standard pseudo-differential operators are constructed. After the authors pass to Toeplitz operators on a closed compact smooth manifold based on pseudo-differential operators and pseudo-differential projections. Further, the authors study elliptic BVPs of Shapiro-Lopatinskii-type in the framework of Boutet de Monvel's calculus. In particular, a \(K\)-theoretic explanation of the existence of such conditions is given and the pseudo-differential structure of parametrices are elucidated. After that, a Toeplitz analogue of the algebras of pseudo-differential operators with the transmission property at the boundary is formulated. The second part studies a Toeplitz analogue of the edge algebra. The authors consider elliptic operators on a manifold with edge, here with edge conditions rather than boundary conditions, again under the unifying goal of understanding Shapiro-Lopatinskii and global projection edge conditions within a Toeplitz analogue of the edge calculus. The third part is devoted to the case of Toeplitz calculus where the underlying space is a smooth manifold with boundary.
scientific article; zbMATH DE number 6967940