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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.
List of contents
Introduction.- Boundary Value Problems with Global Projection Conditions.- Edge Operators with Global Projection Conditions.- BVPs without the Transmission Property.
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"The book is very nice and carefully written, and is a very valuable contribution to the subject ... . It is well suited to experts but also to Ph.D. students who are willing to pursue this beautiful and rich area of research." (Alberto Parmeggiani, Mathematical Reviews, August, 2019)
"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." (David Kapanadze, zbMATH 1423.35003, 2019)
