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Calderón's inverse problem via Vekua theory

Briceyda B. Delgado

TL;DR

The paper addresses Calderón's inverse conductivity problem in dimensions $n\\ge 3$ using a Clifford-analysis framework based on the Vekua equation. It develops generalized Borel-Pompeiu and Green-Vekua integral formulas, linking Vekua solutions to the conductivity and Schrödinger operators and establishing a Clifford-Beltrami equivalence for the main Vekua equation. This provides an alternative route to CGO-based uniqueness proofs via integral representations and boundary data. For Lipschitz conductivities, the authors prove a interior uniqueness result: if the Dirichlet-to-Neumann maps coincide, then the conductivities coincide inside the domain, with an interior identity connecting the potential differences to Schrödinger solutions and a Lipschitz-extension CGO argument reinforcing the result.

Abstract

In this work, we will prove a uniqueness result for Calderón's inverse problem via some integral representation formulas for solutions of the Vekua equation in the framework of Clifford analysis.

Calderón's inverse problem via Vekua theory

TL;DR

The paper addresses Calderón's inverse conductivity problem in dimensions using a Clifford-analysis framework based on the Vekua equation. It develops generalized Borel-Pompeiu and Green-Vekua integral formulas, linking Vekua solutions to the conductivity and Schrödinger operators and establishing a Clifford-Beltrami equivalence for the main Vekua equation. This provides an alternative route to CGO-based uniqueness proofs via integral representations and boundary data. For Lipschitz conductivities, the authors prove a interior uniqueness result: if the Dirichlet-to-Neumann maps coincide, then the conductivities coincide inside the domain, with an interior identity connecting the potential differences to Schrödinger solutions and a Lipschitz-extension CGO argument reinforcing the result.

Abstract

In this work, we will prove a uniqueness result for Calderón's inverse problem via some integral representation formulas for solutions of the Vekua equation in the framework of Clifford analysis.
Paper Structure (7 sections, 15 theorems, 88 equations)

This paper contains 7 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Vekua spaces
  4. Integral representation formulas
  5. Integral formulas for the main Vekua equation
  6. Uniqueness result
  7. Uniqueness result for Lipschitz conductivities

Key Result

Theorem 1

Let $n\geq 3$. Let $f, g\in W^{1,\infty}(\Omega)$ be conductivities away from zero in $\Omega$. If $\Lambda_f=\Lambda_g$, then $f=g$ in $\Omega$.

Theorems & Definitions (22)

  • Theorem 1: Uniqueness result
  • Theorem 2
  • proof
  • Corollary 3
  • Corollary 4
  • proof
  • Corollary 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 12 more