A Local-to-Global Propagation Principle for Dirichlet-to-Neumann Maps
We establish four local-to-global propagation results for Dirichlet--to--Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component \(\Gamma\), then equality of the corresponding local Dirichlet--to--Neumann maps on a nonempty open subset of \(\Gamma\) propagates to equality of the associated global Dirichlet--to--Neumann maps on all of \(\Gamma\). The proof combines unique continuation and self-adjointness arguments. The second replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet--to--Neumann maps. The proof relies on the spectral unique continuation theory of Jerison--Lebeau, through the formulation of Le~Rousseau--Lebeau. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi--analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi--analytic Riemannian manifold, they rely on the propagation theorems of Ganguly--Thangavelu and of Bhowmik--Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg--Marchenko theorem and Weyl--Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi--analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet--to--Neumann maps.