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Layer potential operators for transmission problems on extension domains

Gabriel Claret, Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev

TL;DR

The paper develops a purely variational framework to extend layer potential theory and Neumann–Poincaré analysis to two-sided Sobolev extension domains, including non-Lipschitz and fractal boundaries, without requiring a boundary measure. It defines measure-free trace spaces $\mathcal{B}(\partial\Omega)$ and $\dot{\mathcal{B}}(\partial\Omega)$ and constructs Poincaré–Steklov operators, Neumann–Poincaré operators, and two-sided layer potentials via weak formulations, establishing isometries, bijections, and resolvent representations. It then derives Calderón projectors, energy-norm-based isometries, and spectral-invertibility results, extending the classical Lipschitz theory to rough domains. Finally, it demonstrates imaging applications, including a general representation formula and monotone subdomain identification from boundary data, with special treatment of disk inclusions in 2D. This provides a robust framework for inverse problems on irregular domains where boundary regularity is limited.

Abstract

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{é} operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.

Layer potential operators for transmission problems on extension domains

TL;DR

The paper develops a purely variational framework to extend layer potential theory and Neumann–Poincaré analysis to two-sided Sobolev extension domains, including non-Lipschitz and fractal boundaries, without requiring a boundary measure. It defines measure-free trace spaces and and constructs Poincaré–Steklov operators, Neumann–Poincaré operators, and two-sided layer potentials via weak formulations, establishing isometries, bijections, and resolvent representations. It then derives Calderón projectors, energy-norm-based isometries, and spectral-invertibility results, extending the classical Lipschitz theory to rough domains. Finally, it demonstrates imaging applications, including a general representation formula and monotone subdomain identification from boundary data, with special treatment of disk inclusions in 2D. This provides a robust framework for inverse problems on irregular domains where boundary regularity is limited.

Abstract

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{é} operators. These classes of domains contain Lipschitz domains, and also domains with fractal boundaries. Although our variational formulation does not involve any measures on the boundary, we recover the classical results in smooth domains by considering the surface measure on the boundary. We discuss properties of these operators and generalize basic results in imaging beyond the Lipschitz case.
Paper Structure (24 sections, 50 theorems, 144 equations, 2 figures)

This paper contains 24 sections, 50 theorems, 144 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Boundary value problems on admissible domains
  3. Admissible domains and traces
  4. Orthogonality, harmonic extensions and isometries
  5. Neumann solutions
  6. Normal derivatives
  7. Poincaré-Steklov operators for admissible domains
  8. Transmission problems for admissible domains
  9. Two-sided admissible domains and jumps
  10. Subspaces and orthogonality
  11. Double layer potentials
  12. Single layer potentials
  13. Superposition
  14. Resolvent representations
  15. Representations of single layer potentials
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $\Omega$ be an $H^1$-admissible domain. Then $u\mapsto \operatorname{Tr}_i u$ gives a linear surjection $\operatorname{Tr}_i:H^1(\Omega)\to \mathcal{B}(\partial\Omega)$, well-defined in the sense that given $u\in H^1(\Omega)$, its trace $\operatorname{Tr}_i u$ on $\partial\Omega$ does not depend

Figures (2)

  • Figure 1: Graphic representation of the values of $\lambda$ for which Theorems \ref{['ThSpecNP']} and \ref{['ThSpecNPo']} apply, which are all $\lambda\in\mathbb{C}$ outside the gray area (or on its boundary). The dotted lines correspond to the circles of center $\pm\frac{1}{2}$ and radius $1$.
  • Figure 2: An illustration of the imaging setting with an inclusion $D$ inside a domain $\Omega$, both two-sided $\dot H^1$-admissible. The purpose of the imaging problem is to identify the inclusion $D$ based on measurements on $\partial\Omega$.

Theorems & Definitions (106)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 96 more