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Partial data stability for the inverse fractional conductivity problem

Giovanni Covi, Antti Kujanpää, Jesse Railo

TL;DR

This paper addresses stability in the inverse fractional conductivity problem with partial exterior data. It leverages a fractional Liouville reduction to recast the problem as a fractional Schrödinger equation and then applies established partial-data stability tools, including quantitative unique continuation, to derive sharp stability results. The authors prove two main results: (i) a logarithmic stability bound when the conductivities agree a priori on the exterior, and (ii) a log-log stability bound in $L^p$ when the conductivity difference is compactly supported without exterior agreement. These results extend partial-data stability theory to nonlocal fractional models and quantify how exterior measurements constrain interior conductivities, with explicit modulus of continuity depending on problem data.

Abstract

The classical Calderón problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity problem on bounded smooth domains. Using the fractional Liouville reduction, we obtain a log-log stability estimate when the conductivities a priori agree in the measurement set and their difference has compact support. In the case in which the conductivities are assumed to agree a priori in the whole exterior of the domain, we obtain a shaper logarithmic stability estimate.

Partial data stability for the inverse fractional conductivity problem

TL;DR

This paper addresses stability in the inverse fractional conductivity problem with partial exterior data. It leverages a fractional Liouville reduction to recast the problem as a fractional Schrödinger equation and then applies established partial-data stability tools, including quantitative unique continuation, to derive sharp stability results. The authors prove two main results: (i) a logarithmic stability bound when the conductivities agree a priori on the exterior, and (ii) a log-log stability bound in when the conductivity difference is compactly supported without exterior agreement. These results extend partial-data stability theory to nonlocal fractional models and quantify how exterior measurements constrain interior conductivities, with explicit modulus of continuity depending on problem data.

Abstract

The classical Calderón problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity problem on bounded smooth domains. Using the fractional Liouville reduction, we obtain a log-log stability estimate when the conductivities a priori agree in the measurement set and their difference has compact support. In the case in which the conductivities are assumed to agree a priori in the whole exterior of the domain, we obtain a shaper logarithmic stability estimate.

Paper Structure

This paper contains 9 sections, 14 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the article
  4. Preliminaries
  5. Bessel potential spaces
  6. The fractional Schrödinger equation
  7. The fractional conductivity equation
  8. Partial data problem I
  9. Partial data problem II

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb R^n$ be a bounded, smooth domain, let $0<s<\min(1,n/2)$, and assume that $W_1,W_2 \Subset \Omega_e$ are nonempty open sets. Let $\gamma_1,\gamma_2\in L^\infty({\mathbb R}^n)$ be such that and assume that there exist $\varepsilon, C_1>0$ such that the background deviations $m_1,m_2$ verify Then where $\omega$ is a logarithmic modulus of continuity satisfying for three

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: GSU20
  • Lemma 2.2
  • Lemma 2.3: Liouville reduction, RZ2022LowReg
  • Theorem 3.1: RS-fractional-calderon-low-regularity-stability
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • ...and 12 more