Boundary Value Problems with Global Projection Conditions

Description

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. Front Matter ....Pages i-xvi Front Matter ....Pages 1-1 Pseudo-differential operators (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 3-92 BVPs with the transmission property (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 93-144 Shapiro–Lopatinskii ellipticity (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 145-166 Toeplitz boundary value problems (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 167-178 Cutting and pasting of elliptic operators, Cauchy data spaces (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 179-200 Front Matter ....Pages 201-201 The cone algebra (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 203-284 The edge algebra (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 285-306 Edge-ellipticity (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 307-320 Toeplitz edge problems (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 321-330 Front Matter ....Pages 331-331 The edge approach to BVPs (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 333-348 Boundary ellipticity (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 349-354 Toeplitz boundary value problems without the transmission property (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 355-360 Examples, applications and remarks (Xiaochun Liu, Bert-Wolfgang Schulze)....Pages 361-396 Back Matter ....Pages 397-410