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Anisotropic Calderón Problem for a Non-Local Second Order Elliptic Operator

Susovan Pramanik

TL;DR

This work studies the anisotropic Calderón problem for a non-local second-order elliptic operator $\mathcal{L}_g^{1/2}=((-\,\Delta_g)^2+m^2 I)^{1/2}$ on closed manifolds. It develops a heat-semi-group framework to define fractional powers and analyzes the associated Cauchy data for $\mathcal{A}_g=\mathcal{L}_g^{1/2}-m I$, establishing mapping properties and symbol class for the operator. The core result shows that equality of the Cauchy data on an open set implies the existence of a diffeomorphism $\Phi$ with $\Phi^* g_2 = g_1$, i.e., geometric recovery up to gauge, by linking nonlocal data to the local Calderón-type data via heat-kernel and wave-propagation arguments and a known rigidity theorem. This work extends nonlocal Calderón inverse problems to the critical order $2$ and connects semigroup methods, pseudodifferential calculus, and rigidity results to recover anisotropic geometries from partial boundary measurements, with potential applications in diffusion-related imaging and physics.

Abstract

This paper investigates the anisotropic Calderón problem for a non-local elliptic operator of order 2, on closed Riemannian manifolds. We demonstrate that using the Cauchy data set, we can recover the geometry of a closed Riemannian manifold up to standard gauge.

Anisotropic Calderón Problem for a Non-Local Second Order Elliptic Operator

TL;DR

This work studies the anisotropic Calderón problem for a non-local second-order elliptic operator on closed manifolds. It develops a heat-semi-group framework to define fractional powers and analyzes the associated Cauchy data for , establishing mapping properties and symbol class for the operator. The core result shows that equality of the Cauchy data on an open set implies the existence of a diffeomorphism with , i.e., geometric recovery up to gauge, by linking nonlocal data to the local Calderón-type data via heat-kernel and wave-propagation arguments and a known rigidity theorem. This work extends nonlocal Calderón inverse problems to the critical order and connects semigroup methods, pseudodifferential calculus, and rigidity results to recover anisotropic geometries from partial boundary measurements, with potential applications in diffusion-related imaging and physics.

Abstract

This paper investigates the anisotropic Calderón problem for a non-local elliptic operator of order 2, on closed Riemannian manifolds. We demonstrate that using the Cauchy data set, we can recover the geometry of a closed Riemannian manifold up to standard gauge.
Paper Structure (5 sections, 10 theorems, 97 equations)

This paper contains 5 sections, 10 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminary understandings.
  3. Heat semigroup approach in defining the fractional power of $\mathcal{L}_g$
  4. Symbol and order
  5. Proof of Main Result

Key Result

Theorem 1.1

Let $(M_j, g_j)$ be two smooth, closed, connected Riemannian manifolds, with $M_1 \cap M_2 \neq \emptyset$. Consider $\mathcal{O}_j \subset M_j, \, j = 1, 2$ to be two non-empty open sets such that $(\mathcal{O}_1, g_1) = (\mathcal{O}_2, g_2) =: (\mathcal{O}, g)$. Assume that Then, there exists a diffeomorphism $\Phi: M_1 \to M_2$ such that $\Phi^\ast g_2 = g_1$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1: Fei22
  • Theorem 2.2: FGKRSU25
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 8 more