The local complex Calderón problem. Stability in a layered medium for a special type of anisotropic admittivity

TL;DR

This work addresses the inverse problem of recovering a layered, complex anisotropic admittivity $\sigma=\gamma A$ from boundary measurements by focusing on a piecewise affine $\gamma$ and a known anisotropy $A$. By formulating the forward problem as a 2×2 real system and employing an augmented domain with Green functions, the authors derive a Lipschitz stability bound $\|\sigma^{(1)}-\sigma^{(2)}\|_{L^{\infty}(\Omega)} \le C\|\Lambda_1^{\Sigma}-\Lambda_2^{\Sigma}\|_*$, under robust a-priori assumptions on domain geometry and material properties. The analysis rests on sharp Green function asymptotics, Alessandrini-type identities, and quantitative unique continuation across layer interfaces, enabling stable reconstruction in a layered, anisotropic, and complex-valued setting. Additionally, the paper introduces a misfit functional framework yielding Hölder stability with respect to the misfit, bridging theoretical results with practical, locally measured data and potential numerical implementations.

Abstract

We deal with Calderón's problem in a layered anisotropic medium $Ω\subset\mathbb{R}^n$, $n\geq 3$, with complex anisotropic admittivity $σ=γA$, where $A$ is a known Lipschitz matrix-valued function. We assume that the layers of $Ω$ are fixed and known and that $γ$ is an unknown affine complex-valued function on each layer. We provide Hölder and Lipschitz stability estimates of $σ$ in terms of an ad hoc misfit functional as well as the more classical Dirichlet to Neumann map localised on some open portion $Σ$ of $\partialΩ$, respectively.

Theorems & Definitions (20)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.3
• Definition 3.1
• Theorem 3.1
• proof