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Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains

Gabriel Claret

TL;DR

This work develops a measure-free boundary integral framework for the Helmholtz transmission problem on extension domains, including fractal and varying-dimension boundaries. It generalizes layer-potential theory, Neumann–Poincaré operators, and Calderón projectors to these domains, enabling a robust link between two-sided transmission problems and one-sided scattering problems with Dirichlet, Neumann, and Robin (impedance) boundary conditions treated via trace-space duality. The authors establish well-posedness, derive boundary-integral equations, and provide a representation of solutions through Newtonian potentials, while extending impedance-only scattering analysis to extension-domain obstacles in any dimension. They also prove the continuity of far-field power with respect to impedance and demonstrate the existence of optimal impedances within compact sets, yielding practical implications for impedance design in irregular geometries. Overall, the paper broadens the applicability of boundary-integral methods to complex geometries and provides a pathway for impedance-optimization in advanced acoustic scattering settings.

Abstract

We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension.

Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains

TL;DR

This work develops a measure-free boundary integral framework for the Helmholtz transmission problem on extension domains, including fractal and varying-dimension boundaries. It generalizes layer-potential theory, Neumann–Poincaré operators, and Calderón projectors to these domains, enabling a robust link between two-sided transmission problems and one-sided scattering problems with Dirichlet, Neumann, and Robin (impedance) boundary conditions treated via trace-space duality. The authors establish well-posedness, derive boundary-integral equations, and provide a representation of solutions through Newtonian potentials, while extending impedance-only scattering analysis to extension-domain obstacles in any dimension. They also prove the continuity of far-field power with respect to impedance and demonstrate the existence of optimal impedances within compact sets, yielding practical implications for impedance design in irregular geometries. Overall, the paper broadens the applicability of boundary-integral methods to complex geometries and provides a pathway for impedance-optimization in advanced acoustic scattering settings.

Abstract

We consider a transmission problem for the Helmholtz equation across the boundary of an extension domain. A such boundary can be Lipschitz, fractal, or of varying Hausdorff dimension for instance. We generalise the notions of layer potential and Neumann-Poincar{é} operators, and of Calder{ó}n projectors in that context. Those boundary operators allow to connect the transmission problem (on the whole space) to one-sided problems -- notably, scattering problems -- with Dirichlet, Neumann and Robin boundary conditions. Since an extension domain needs no specific boundary measure, the Robin (impedance) condition is not understood in a boundary L^2-type space, rather by duality on the trace space itself. We discuss the well-posedness of the impedance scattering problem in that framework and compare to the classical L^2 setting. Our analysis allows to generalise optimisation results for acoustic scattering when the obstacle is an extension domain in any dimension.
Paper Structure (13 sections, 24 theorems, 116 equations, 2 figures)

This paper contains 13 sections, 24 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Functional framework
  3. Helmholtz transmission problem
  4. Layer potential operators
  5. Single layer potential
  6. Double layer potential
  7. Representation of the layer potential operators
  8. Boundary values of the layer potentials
  9. Neumann-Poincaré operators
  10. Calderón projectors
  11. Scattering problem
  12. Intrinsic impedance boundary condition
  13. Impedance optimisation

Key Result

Theorem 2.3

Let $\Omega$ be an admissible domain of $\mathbb{R}^n$. Then, the following assertions hold. If $\overline{\Omega}^c$ is admissible, then counterparts of (i), (ii) and (iii) hold for $\operatorname{Tr}^e_{\partial\Omega}$ instead of $\operatorname{Tr}^i_{\partial\Omega}$, replacing $\Omega$ with $\overline{\Omega}^c$. In particular, the norm is a Hilbert space norm on $\mathcal{B}(\partial\Omega

Figures (2)

  • Figure 1: On the left, an example of a two-sided admissible domain of $\mathbb{R}^2$. Its boundary is composed of parts of different Hausdorff dimensions: three smooth lines of dimension $1$, a Koch curve of dimension $\frac{\ln 4}{\ln 3}$, a Minkowski curve of dimension $\frac{\ln 8}{\ln 4}$ and a pentagon-inspired fractal curve of dimension $\frac{\ln 14}{\ln 5}$. On the right, a domain of $\mathbb{R}^2$ which is not an extension domain due to the presence of cusps. The presence of ingoing cusps also prevents the complement domain from being an extension domain.
  • Figure 2: Illustration of the setting of the optimisation problem.

Theorems & Definitions (51)

  • Definition 2.1: Admissible domain
  • Definition 2.2: Trace operators
  • Theorem 2.3: Trace theorem
  • Remark 2.4
  • Definition 2.5: Weak normal derivatives
  • Theorem 2.6: Green's formula
  • proof
  • Lemma 2.7
  • proof
  • Proposition 2.8: claret_layer_2025
  • ...and 41 more