English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We investigate a linearised Calder\'on problem in a two-dimensional boundedsimply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearisedproblem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega)= \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each ofthe infinite-dimensional $\mathcal{H}_k$ subspaces. In particular,$\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. Thisappears to be the first Lipschitz stability result for infinite-dimensionalspaces of perturbations in the context of the (linearised) Calder\'on problem.Previous optimal estimates with respect to the operator norm of the data maphave been of the logarithmic-type in infinite-dimensional settings. Theremarkable improvement is enabled by using the Hilbert-Schmidt norm for theNeumann-to-Dirichlet boundary map and its Fr\'echet derivative with respect tothe conductivity coefficient. We also derive a direct reconstruction methodthat 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.

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