Density of Products of Solutions and the Linearized Calderón Problem in an Analytic Background
Abstract
We study the linearization, with respect to zeroth-order perturbations, of the local Dirichlet-to-Neumann map of a formally self-adjoint elliptic operator P=−Δg+Q on a compact Riemannian manifold (M,g) of dimension n≥ 2 with boundary, where Q is of order at most one. Near a boundary point where the boundary, metric, and P are analytic, we show that the linearized map is a classical analytic pseudodifferential operator of order −1 if and only if the zeroth-order perturbation q extends analytically across the boundary. We also prove that local boundary data in an arbitrary open analytic boundary patch determines q locally. In particular, if the measurement set meets every connected component of M, the local linearized DN map is injective on L∞ perturbations. Finally, we obtain a global completeness result for products of pairs of solutions whose boundary traces are supported in the measurement set. The global recovery proceeds by local steps using Runge approximation.
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BibTeX
@misc{busch2026density,
title = {{Density of Products of Solutions and the Linearized Calderón Problem in an Analytic Background}},
author = {Leonard Busch and Tony Liimatainen and Gunther Uhlmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19288},
eprint = {2610.19288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19288},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Density of Products of Solutions and the Linearized Calderón Problem in an Analytic Background
AU - Leonard Busch
AU - Tony Liimatainen
AU - Gunther Uhlmann
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19288
UR - https://arxiv.org/abs/2610.19288
AB - We study the linearization, with respect to zeroth-order perturbations, of the local Dirichlet-to-Neumann map of a formally self-adjoint elliptic operator P=−Δ_g+Q on a compact Riemannian manifold (M,g) of dimension n≥ 2 with boundary, where Q is of order at most one. Near a boundary point where the boundary, metric, and P are analytic, we show that the linearized map is a classical analytic pseudodifferential operator of order −1 if and only if the zeroth-order perturbation q extends analytically across the boundary. We also prove that local boundary data in an arbitrary open analytic boundary patch determines q locally. In particular, if the measurement set meets every connected component of M, the local linearized DN map is injective on L^∞ perturbations. Finally, we obtain a global completeness result for products of pairs of solutions whose boundary traces are supported in the measurement set. The global recovery proceeds by local steps using Runge approximation.
ER -