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The Calderón Problem for Quasilinear Conductivities of Conformally Transversally Anisotropic Media

Xi Chen, Ziyun Jin

TL;DR

This work extends Calderón’s inverse boundary value problem to quasilinear anisotropic conductivities on conformally transversally anisotropic (CTA) manifolds. It develops a higher-order linearization framework combined with complex geometric optics (CGO) and Gaussian beam constructions on CTA geometries to recover the derivatives $\partial_{(s,p)}^{\beta} a(s,x,p)$ at $(s,p)=(t,0)$ from boundary measurements; in particular, all derivatives are determined, and if $a$ is analytic in $p$, the full nonlinear conductivity is uniquely recovered. The analysis reduces the nonlinear inverse problem to linear inverse problems and leverages CTA structure to obtain uniqueness via CGO/Gaussian beams, including forward problem well-posedness and gauge invariance. Overall, the paper broadens Calderón-type identifiability to nonlinear, anisotropic media in CTA settings, providing theoretical foundations for boundary-based reconstruction in more complex EIT-like models.

Abstract

This paper investigates Calderón's problem on a conformally transversally anisotropic manifold $ (M,g) $ of dimension $n \geq 3$, where the conductivity $ a(s,x,p) $ might depend on both the electric potential and the electric field. We establish that for all $(t,x)\in \mathbb{R}\times M$ and $β\in \mathbb{N}^{1+n}$ the derivatives $ \partial_{(s,p)}^βa(s,x,p)|_{(s,p)=(t,0)}$ are uniquely determined by the boundary voltage-current measurements. If $ a(s,x,p) $ is analytic in $ p $, then $ a(s,x,p) $ can be uniquely recovered.

The Calderón Problem for Quasilinear Conductivities of Conformally Transversally Anisotropic Media

TL;DR

This work extends Calderón’s inverse boundary value problem to quasilinear anisotropic conductivities on conformally transversally anisotropic (CTA) manifolds. It develops a higher-order linearization framework combined with complex geometric optics (CGO) and Gaussian beam constructions on CTA geometries to recover the derivatives at from boundary measurements; in particular, all derivatives are determined, and if is analytic in , the full nonlinear conductivity is uniquely recovered. The analysis reduces the nonlinear inverse problem to linear inverse problems and leverages CTA structure to obtain uniqueness via CGO/Gaussian beams, including forward problem well-posedness and gauge invariance. Overall, the paper broadens Calderón-type identifiability to nonlinear, anisotropic media in CTA settings, providing theoretical foundations for boundary-based reconstruction in more complex EIT-like models.

Abstract

This paper investigates Calderón's problem on a conformally transversally anisotropic manifold of dimension , where the conductivity might depend on both the electric potential and the electric field. We establish that for all and the derivatives are uniquely determined by the boundary voltage-current measurements. If is analytic in , then can be uniquely recovered.

Paper Structure

This paper contains 11 sections, 15 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Forward problems
  4. The Calderón problem for linear conductivities
  5. Linearization of Nonlinear Equations
  6. Recover the Lower-Order Derivatives
  7. First-Order Linearization: Recover the Conformal Factor
  8. Second-Order Linearization: Recover the First Derivatives
  9. Recover the Higher-Order Derivatives
  10. Third-Order Linearization: Constructing CGO Solutions
  11. Higher-Order Linearization: Proof by Induction on Orders

Key Result

Theorem 1

Let $(M,\partial M, g)$ be a compact simple CTA manifold with smooth boundary $\partial M$. Write $\gamma_m(s,x,p):= a_m(s,x,p)A(x)$, for $m=1,2$ as in (def-form) and (def:corresponding_metric). If $N_{\gamma_1}=N_{\gamma_2}$, then for all $(t,x)\in\mathbb{R}\times M$ and $\beta\in\mathbb{N}^{1+n}$, If in addition, $a_m(s,x,p)$ is analytic in $p$, then $a_1=a_2$.

Theorems & Definitions (28)

  • Definition 1: Gauge transform
  • Definition 2: CTA manifold
  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['lmm:gauge-invariance']}
  • Proposition 1
  • Lemma 3
  • Proposition 2
  • ...and 18 more