Transfer of Stability from the Classical to the Fractional Anisotropic Calderón Problem

TL;DR

The paper tackles the problem of translating stability from local Calderón inverse problems to their nonlocal fractional counterparts in two settings: (i) a fractional Calderón problem on closed smooth manifolds and (ii) the spectral Dirichlet fractional Laplacian on bounded Lipschitz domains. It develops a unified framework based on the Caffarelli-Silvestre extension and quantitative unique continuation, enabling a quantitative transfer of uniqueness and, under a-priori bounds, a logarithmic-stability transfer from local to nonlocal data via source-to-solution operators. In the spectral setting, it proves qualitative and quantitative transfer of uniqueness from local to nonlocal problems, including isotropic stability results, and it establishes a quantitative relation between local StoS and Dirichlet-to-Neumann measurements for the classical Calderón problem. The work further provides a local-to-nonlocal stability transfer by combining Runge approximation with Alessandrini identities, and discusses optimality through instability bounds, highlighting that the principal-part stability in fractional Calderón problems is, in general, at best logarithmic. Overall, the results offer the first rigorous, quantitative bridge between local and nonlocal Calderón problems in these settings, with implications for stability analyses in anisotropic and spectral frameworks.

Abstract

We discuss two spectral fractional anisotropic Calderón problems with source-to-solution measurements and their quantitative relation to the classical Calderón problem. Firstly, we consider the anistropic fractional Calderón problem from [FGKU25]. In this setting, we quantify the relation between the local and nonlocal Calderón problems which had been deduced in [R25] and provide an associated stability estimate. As a consequence, any stability result which holds on the level of the local problem with source-to-solution data has a direct nonlocal analogue (up to a logarithmic loss). Secondly, we introduce and discuss the fractional Calderón problem with source-to-solution measurements for the spectral fractional Dirichlet Laplacian on open, bounded, connected, Lipschitz sets on $\mathbb{R}^n$. Also in this context, we provide a qualitative and quantitative transfer of uniqueness from the local to the nonlocal setting. As a consequence, we infer the first stability results for the principal part for a fractional Calderón type problem for which no reduction of Liouville type is known. Our arguments rely on quantitative unique continuation arguments. As a result of independent interest, we also prove a quantitative relation between source-to-solution and Dirichlet-to-Neumann measurements for the classical Calderón problem.

Figures (4)

• Figure 1: Schematic set-up for our formulation of the (fractional) Calderón problem for the spectral (fractional) Laplacian.
• Figure 2: Schematic visualization of Step 2 in the proof of Proposition \ref{['prop:stability_estimate']}. We have smallness of the data on the boundary $O \times \{0\}$ (violet part). By Proposition \ref{['prop:bbucp']} we transfer the smallness into the bulk, more precisely, to the $(n+1)$-dimensional half-balls $B_{c\bar{r}}^+(x_j)$ (green half-balls). Then by an iterative application of Proposition \ref{['prop:3ballsinequ']} we are able to propagate the smallness upwards in the bulk along a sequence of balls (gray balls) to a certain cut-off height $L$. For visualization, sizes and positions of balls are not true-to-scale.
• Figure 3: More precise visualization of the set-up of balls in Step 2 of the proof of Proposition \ref{['prop:stability_estimate']}. The green coloured half-balls represent the $(n+1)$-dimensional half-balls $B_{c\bar{r}}^+(x_j)$ and the gray coloured balls are the $(n+1)$-dimensional balls $B_{2r}(x_i^{(k)})$. For visualization, sizes or positions of the balls may not be perfectly true-to-scale.
• Figure 4: Schematic set-up in the proof of the quantitative Runge approximation, Proposition \ref{['prop:quantitative_Runge_approx']}. Here, $A \Subset A_+ \Subset \Omega$ and $\Gamma$ is an open neighborhood of $\partial A_+$ with $\Gamma \Subset \Omega \setminus \overline{A}$.

Theorems & Definitions (59)

• Theorem 1.1: Transfer of single measurement stability
• Theorem 1.2: Transfer of infinite measurement stability
• Corollary 1.3: Conditional transfer of stability: from local estimates to nonlocal ones
• Theorem 1.4: Density
• Remark 1.5
• Corollary 1.6: Uniqueness
• Theorem 1.7: Transfer of stability in the single measurement setting
• Corollary 1.8: Stability for isotropic metrics
• Proposition 1.9
• Remark 1.10