Calderón's inverse problem via Vekua theory

Technical Summary: Calderón's Inverse Problem via Vekua Theory

Problem Statement
This work addresses the Calderón inverse conductivity problem in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$) with a Lipschitz boundary. The objective is to recover a non-vanishing conductivity function $\sigma \in W^{1,\infty}(\Omega)$ from the Dirichlet-to-Neumann (DtN) map $\Lambda_\sigma$. The DtN map is defined by the boundary values of the solution $u_0$ to the conductivity equation $\text{div}(\sigma \nabla u_0) = 0$, specifically mapping the Dirichlet data $\phi_0$ to the Neumann data $\sigma \nabla u_0 \cdot \eta$. The central question is whether the map $\sigma \mapsto \Lambda_\sigma$ is injective, i.e., whether $\Lambda_f = \Lambda_g$ implies $f = g$ in $\Omega$.

Methodology
The paper employs the framework of Clifford analysis to analyze the space of solutions to the Vekua equation. The methodology proceeds through the following steps:

1. Clifford Analysis and Vekua Equations: The authors utilize the real Clifford algebra $Cl_{0,n}$ and the Moisil-Teodorescu operator $D$. They study the generalized Vekua equation $Dw = \alpha w$ (and its conjugate variants), where $\alpha$ is related to the conductivity. Specifically, for the "main" Vekua equation, $\alpha = \nabla f / f$, where $f$ is the conductivity.
2. Hodge Decomposition: Building on previous work [16], the paper utilizes a Hodge decomposition for generalized Vekua spaces. This establishes an orthogonal decomposition of the Hilbert space $L^2(\Omega, Cl_{0,n})$ into the space of solutions to the Vekua equation and the image of a specific operator involving the Teodorescu transform.
3. Integral Representation Formulas: A core technical contribution is the derivation of generalized Borel-Pompeiu and partial Cauchy integral formulas. These formulas relate the scalar part of a Vekua solution to boundary integrals involving fundamental solutions of the associated operators ($D - M_{\alpha}C$ and $D - \alpha C$).
4. Connection to Schrödinger and Conductivity Equations: The paper establishes a factorization linking the main Vekua operator to the Schrödinger operator $-\Delta + q$, where the potential $q = \Delta f / f$. This allows the scalar part of the Vekua solution to satisfy the conductivity equation.
5. Complex Geometrical Optics (CGO) Solutions: For the uniqueness proof, the authors adapt the construction of CGO solutions (originally by Pöivärinta, Panchenko, and Uhlmann [26]) for conductivities in $W^{1,\infty}$. These solutions are of the form $u = e^{x \cdot \zeta}(f^{-1} + \psi)$, where $\psi$ decays in a weighted Sobolev space.

Key Contributions and Results

• Integral Formulas for Vekua Solutions: The paper derives integral representation formulas (Theorem 2, Corollary 3) that recover the scalar part of Vekua solutions from their boundary values. These formulas generalize classical Cauchy formulas to the Clifford setting and explicitly incorporate the DtN maps associated with both the conductivity and Schrödinger equations.
• Generalized Green-Vekua Identity: Proposition 12 presents a generalized identity relating the difference of potentials ($\Delta g/g - \Delta f/f$) to the difference of solutions of the associated Schrödinger equations. This identity serves as a bridge between the boundary measurements and the internal properties of the conductivities.
• Uniqueness for $W^{2,\infty}$ and $W^{1,\infty}$ Conductivities:
  • For conductivities in $W^{2,\infty}(\Omega)$, the paper provides an alternative proof of uniqueness by showing that if $\Lambda_f = \Lambda_g$, then the potentials $\Delta f/f$ and $\Delta g/g$ are equal, which implies $f=g$ given the boundary regularity (Lemma 11).
  • For the more general case of Lipschitz conductivities ($W^{1,\infty}(\Omega)$), the authors prove that the DtN map uniquely determines the solutions to the conductivity equation within the domain (Proposition 14).
• Main Theorem (Theorem 1): The paper proves that for $n \geq 3$, if $f, g \in W^{1,\infty}(\Omega)$ are conductivities bounded away from zero and $\Lambda_f = \Lambda_g$, then $f = g$ in $\Omega$. The proof relies on the equivalence of solutions derived from the integral formulas and the asymptotic behavior of CGO solutions as $|\zeta| \to \infty$.

Significance and Claims
The paper claims to offer an alternative perspective on the uniqueness of the Calderón problem by utilizing Vekua theory within Clifford analysis. Rather than relying solely on the traditional reduction to the Schrödinger equation via the Liouville transformation, the authors demonstrate how integral representation formulas for Vekua solutions can directly yield the necessary identities to prove uniqueness.

The authors state that their methods provide:

1. An alternative proof of well-known uniqueness results for conductivities in $W^{2,\infty}$ and $W^{1,\infty}$.
2. A framework that may be adaptable to related inverse problems, such as the anisotropic Maxwell equation.
3. A demonstration that the unique extensions of solutions in $H^1(\Omega)$ are determined by boundary measurements, complementing existing identifiability results regarding complex geometrical solutions in the exterior domain.

The work does not propose new experimental applications or future extensions beyond the mathematical framework of inverse problems and Clifford analysis, maintaining a focus on theoretical rigor and the development of analytical tools for inverse conductivity problems.