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Poincar{é}-Steklov operator and Calder{ó}n's problem on extension domains

Gabriel Claret, Michael Hinz, Anna Rozanova-Pierrat

TL;DR

This work extends Calderón's inverse problem to $H^1$-extension domains that may possess non-smooth or fractal boundaries by broadening the Dirichlet-to-Neumann (Poincaré-Steklov) map to a capacity-based trace framework. It establishes direct problem stability for bounded conductivities continuous near the boundary and develops a comprehensive inverse problem theory: boundary stability for Lipschitz conductivities, domain identification under Lipschitz regularity, and domain stability for $W^{2,\infty}$ conductivities constant near the boundary, all in dimensions $n\ge3$. The analysis leverages the conductivity–Schrödinger equivalence and complex geometrical optics (CGO) solutions, together with an exterior corkscrew condition to handle irregular boundaries. The results provide quantitative stability estimates and identification results, enabling boundary measurements to recover internal conductivities on complex, non-Lipschitz domains relevant to applications with fractal-like geometries.

Abstract

We consider Calder{ó}n's problem on a class of Sobolev extension domains containing non-Lipschitz and fractal shapes. We generalize the notion of Poincar{é}-Steklov (Dirichlet-to-Neumann) operator for the conductivity problem on such domains. From there, we prove the stability of the direct problem for bounded conductivities continuous near the boundary. Then, we turn to the inverse problem and prove its stability at the boundary for Lipschitz conductivities, which we use to identify such conductivities on the domain from the knowledge of the Poincar{é}-Steklov operator. Finally, we prove the stability of the inverse problem on the domain for W^{2,$\infty$} conductivities constant near the boundary. The last two results are valid in dimension n $\ge$ 3.

Poincar{é}-Steklov operator and Calder{ó}n's problem on extension domains

TL;DR

This work extends Calderón's inverse problem to -extension domains that may possess non-smooth or fractal boundaries by broadening the Dirichlet-to-Neumann (Poincaré-Steklov) map to a capacity-based trace framework. It establishes direct problem stability for bounded conductivities continuous near the boundary and develops a comprehensive inverse problem theory: boundary stability for Lipschitz conductivities, domain identification under Lipschitz regularity, and domain stability for conductivities constant near the boundary, all in dimensions . The analysis leverages the conductivity–Schrödinger equivalence and complex geometrical optics (CGO) solutions, together with an exterior corkscrew condition to handle irregular boundaries. The results provide quantitative stability estimates and identification results, enabling boundary measurements to recover internal conductivities on complex, non-Lipschitz domains relevant to applications with fractal-like geometries.

Abstract

We consider Calder{ó}n's problem on a class of Sobolev extension domains containing non-Lipschitz and fractal shapes. We generalize the notion of Poincar{é}-Steklov (Dirichlet-to-Neumann) operator for the conductivity problem on such domains. From there, we prove the stability of the direct problem for bounded conductivities continuous near the boundary. Then, we turn to the inverse problem and prove its stability at the boundary for Lipschitz conductivities, which we use to identify such conductivities on the domain from the knowledge of the Poincar{é}-Steklov operator. Finally, we prove the stability of the inverse problem on the domain for W^{2,} conductivities constant near the boundary. The last two results are valid in dimension n 3.
Paper Structure (14 sections, 21 theorems, 100 equations, 1 figure)

This paper contains 14 sections, 21 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Functional framework
  3. Admissible domains
  4. Poincaré-Steklov operator for conductivity problem
  5. Calderón's problem on extension domains
  6. Direct problem
  7. Inverse problem
  8. Stability at the boundary
  9. Identification on the domain
  10. Stability on the domain
  11. Well-posedness and a priori estimates
  12. Laplace equation
  13. Conductivity equation
  14. Proofs of Subsection \ref{['Subsec:Direct-Problem']}

Key Result

Theorem 2.3

Let $\Omega$ be an admissible domain of $\mathbb{R}^n$. Then, the following assertions hold.

Figures (1)

  • Figure 1: On the left, an admissible domain of $\mathbb{R}^2$. The top part of its boundary consists of a Von Koch curve of Hausdorff dimension $\frac{\ln 4}{\ln 3}$, while the bottom part is of Hausdorff dimension $1$. On the right, a domain of $\mathbb{R}^2$ which is not an extension domain for it has outward cusps.

Theorems & Definitions (45)

  • Definition 2.1: Admissible domains
  • Definition 2.2: Trace operator
  • Theorem 2.3: Trace theorem
  • proof
  • Definition 2.4: Weak normal derivative
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7: Poincaré-Steklov operator
  • Proposition 2.8
  • ...and 35 more