Continuity of family of Calder\'on projections
Abstract
We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calder\'on projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces; elliptic regularity; Green's formula and trace theorems for Sobolev spaces; well-posed boundary conditions; duality of spaces and operators in Hilbert space; and the interpolation theorem for operators in Sobolev spaces. Calder\'on projection Cauchy data spaces Elliptic differential operators Green's formula Interpolation theorem Manifolds with boundary Parameter dependence Trace theorem Variational properties