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The Calderón projection: New definition and applications. (English) Zbl 1221.58016
Let \(A\) be an arbitrary linear elliptic first-order differential operator with smooth coefficients acting between sections of complex vector bundles \(E,F\) over a compact smooth manifold \(M\) with smooth boundary \(\Sigma.\) The authors describe the analytic and topological properties of \(A\) in a collar neighborhood \(U\) of \(\Sigma \) and analyze various ways of writing \(A \upharpoonright U\) in product form. They discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of \(A\) by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderón projection. The construction is applied to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderón projection and of well-posed self-adjoint Fredholm extensions under continuous variation of the data.

MSC:
58J32 Boundary value problems on manifolds
35J67 Boundary values of solutions to elliptic equations and elliptic systems
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
57Q20 Cobordism in PL-topology
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