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Convergence of layer potentials and Riemann-Hilbert problem on extension domains

Gabriel Claret, Anna Rozanova-Pierrat, Alexander Teplyaev

Abstract

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{é} operators, Calder{ó}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

Convergence of layer potentials and Riemann-Hilbert problem on extension domains

Abstract

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{é} operators, Calder{ó}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

Paper Structure

This paper contains 16 sections, 50 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dyadic approximation of open sets
  3. Functional framework
  4. Admissible domains, extensions and layer potentials
  5. Convergence along sequences of Hilbert spaces
  6. Convergence of layer potential operators for transmission problems
  7. Convergence on interior domains
  8. Convergence on exterior domains
  9. Convergence on the whole space
  10. Convergence of Neumann-Poincaré operators and Calderón projectors
  11. Convergence of Neumann series for the Neumann-Poincaré operators
  12. Application to the Riemann-Hilbert problem on admissible domains
  13. Cauchy integral, Hilbert transform and layer potentials for Lipschitz domains
  14. Cauchy integral and Hilbert transform for two-sided admissible domain
  15. Equivalence of trace norms
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\Omega$ be an arbitrary open set in $\mathbb{R}^n$. For any $(\pi_m)_{m\in M}$ as in Eq:Root, denoting by $(\Omega^{\square}_k)_{k\in\mathbb{N}}$ the dyadic approximations of $\Omega$ rooted in $(\pi_m)$, it holds: In addition, if $B$ is an open ball, then

Figures (3)

  • Figure 1: Dyadic approximations of a Von Koch snowflake $\Omega$ (lying inside $\partial\Omega$, in blue) in $\mathbb{R}^2$ and of its complementary open set $\overline{\Omega}^c$. The dyadic approximation of $\Omega$ rooted at $\pi_1$, $\Omega^{\square}_k$, lies inside the green dashed line. The dyadic approximation of $\overline{\Omega}^c$ rooted at $\pi_2$, $(\overline{\Omega}^c)^{\square}_k$, lies outside the red dotted line.
  • Figure 2: On the left, a two-sided 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 $\mathbb{R}^3$ which is not an extension domain for it has an outward cusp.
  • Figure 3: Illustration of the impact the sequence of linear maps has on the convergences from Definitions \ref{['Def:CV-Hilb']} and \ref{['Def:CV-Vect']}. The sequence $(H_k)_{k\in\mathbb{N}}$ converges to $H$ through $(H,(\mathcal{T}_k)_{k\in\mathbb{N}})$ and through $(H,(\mathcal{R}_k)_{k\in\mathbb{N}})$. The ellipses represent the spheres of center $0$ and radius $\lVert u\rVert_H$ with respect to the norm on each space. Since the sequences of operators $(\mathcal{T}_k)_{k\in\mathbb{N}}$ and $(\mathcal{R}_k)_{k\in\mathbb{N}}$ are not asymptotically close (in the sense of Lemma \ref{['Lem:EquivCVFramework']}), they induce different notions of convergence: the sequence $(u_k\in H_k)_{k\in\mathbb{N}}$ goes to $u\in H$ through $(\mathcal{T}_k)_{k\in\mathbb{N}}$, but not through $(\mathcal{R}_k)_{k\in\mathbb{N}}$.

Theorems & Definitions (108)

  • Theorem 2.1: Convergence of the dyadic approximations
  • proof
  • Definition 3.1: Admissible domain
  • Definition 3.2: Trace operators
  • Theorem 3.3: Trace theorem
  • Proposition 3.4
  • proof
  • Definition 3.5: Weak normal derivatives
  • Proposition 3.6: claret_layer_2025
  • Definition 3.7: Layer potential operators claret_layer_2025
  • ...and 98 more