Linearised Calder\'on problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations

TL;DR

This paper establishes Lipschitz stability for the linearised Calderón problem in infinite-dimensional spaces of unbounded perturbations, using Hilbert-Schmidt norms to improve upon previous logarithmic estimates and providing a direct reconstruction method.

Contribution

It introduces the first Lipschitz stability result for infinite-dimensional perturbation spaces in the linearised Calderón problem, utilizing Hilbert-Schmidt norms for improved stability estimates.

Findings

Lipschitz stability proven on each subspace of $L^2( abla)$ perturbations.

First stability result of its kind for infinite-dimensional spaces in this context.

Provides a reconstruction method for orthogonal projections of perturbations.

Abstract

We investigate a linearised Calder\'on problem in a two-dimensional bounded simply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearised problem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calder\'on problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fr\'echet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(\Omega)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(\Omega)$ perturbation.