Continuity of family of Calder\'on projections

TL;DR

This paper proves the continuity of Calderón projections associated with a continuous family of elliptic differential operators on manifolds with boundary, using elementary functional analysis tools.

Contribution

It establishes the continuity of Calderón projections and Cauchy data spaces for families of elliptic operators, assuming constant inner solution dimension, with elementary methods.

Findings

Calderón projections form a continuous family of projections.

Cauchy data spaces vary continuously in Sobolev space topology.

Results rely on classical elliptic regularity and operator theory.

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. \keywords{Calder{\'o}n projection\and Cauchy data spaces \and Elliptic differential operators \and Green's formula\and Interpolation theorem\and Manifolds with boundary\and Parameter dependence \and Trace theorem \and Variational properties