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Limiting absorption principle for a hybrid resonance in a two-dimensional cold plasma

Maryna Kachanovska, Étienne Peillon

TL;DR

The paper proves a rigorous limiting absorption principle for a 2D boundary-value problem modeling hybrid plasma resonance, where the principal part vanishes on a curve and changes sign across it. By introducing weighted Sobolev spaces and decomposing solutions into regular and singular parts, the authors obtain uniform stability as the absorption parameter $\nu\to 0^+$ and establish a precise logarithmic singularity along the interface. A key innovation is a radiation-like condition linking the Dirichlet and Neumann traces, which selects a unique LAP limit $u^+$ and yields a well-posed limiting problem. The framework is extended from a simplified rectangle to more general domains, including holes, and is supported by sophisticated tools: extension operators, Bessel potentials, frequency filters, and tailored Green's formulas for singular functions. The results provide a robust analytic basis for hybrid resonance phenomena in cold plasmas with matrix-valued material tensors and have potential relevance for numerical methods and PDE models of plasma dynamics.

Abstract

We study a limiting absorption principle for the boundary-value problem describing a hybrid plasma resonance, with a regular coefficient in the principal part of the operator that vanishes on a curve inside the domain and changes its sign across this curve. We prove the limiting absorption principle by establishing a priori bounds on the solution in certain weighted Sobolev spaces. Next, we show that the solution can be decomposed into regular and singular parts. A peculiar property of this decomposition enables us to introduce a radiation-like condition in a bounded domain and to state a well-posed problem satisfied by the limiting absorption solution.

Limiting absorption principle for a hybrid resonance in a two-dimensional cold plasma

TL;DR

The paper proves a rigorous limiting absorption principle for a 2D boundary-value problem modeling hybrid plasma resonance, where the principal part vanishes on a curve and changes sign across it. By introducing weighted Sobolev spaces and decomposing solutions into regular and singular parts, the authors obtain uniform stability as the absorption parameter and establish a precise logarithmic singularity along the interface. A key innovation is a radiation-like condition linking the Dirichlet and Neumann traces, which selects a unique LAP limit and yields a well-posed limiting problem. The framework is extended from a simplified rectangle to more general domains, including holes, and is supported by sophisticated tools: extension operators, Bessel potentials, frequency filters, and tailored Green's formulas for singular functions. The results provide a robust analytic basis for hybrid resonance phenomena in cold plasmas with matrix-valued material tensors and have potential relevance for numerical methods and PDE models of plasma dynamics.

Abstract

We study a limiting absorption principle for the boundary-value problem describing a hybrid plasma resonance, with a regular coefficient in the principal part of the operator that vanishes on a curve inside the domain and changes its sign across this curve. We prove the limiting absorption principle by establishing a priori bounds on the solution in certain weighted Sobolev spaces. Next, we show that the solution can be decomposed into regular and singular parts. A peculiar property of this decomposition enables us to introduce a radiation-like condition in a bounded domain and to state a well-posed problem satisfied by the limiting absorption solution.
Paper Structure (72 sections, 81 theorems, 436 equations, 2 figures)

This paper contains 72 sections, 81 theorems, 436 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A simplified problem, notation, principal results
  3. A simplified problem on a rectangle
  4. The geometry and the boundary-value problem
  5. Notation
  6. Sobolev spaces with periodic and homogeneous boundary conditions
  7. Standard trace operators and their restrictions
  8. A preliminary well-posedness result and motivation
  9. Principal results for the simplified problem
  10. A road-map to the proofs of the results of the paper
  11. Problems in sub-domains
  12. A homogeneous Neumann problem
  13. Regular problem \ref{['eq:rp']}: well-posedness in $\mathcal{V}_{reg}(\Omega_p)$
  14. Auxiliary results needed to prove Theorem \ref{['theorem:well_posedness_regular']}
  15. Proof of Theorem \ref{['theorem:well_posedness_regular']}
  16. ...and 57 more sections

Key Result

Corollary 2.2

The matrix-valued function $\mathbf{x}\mapsto\mathbb{M}_{\nu}(\mathbf{x}):=x\mathbb{A}(\mathbf{x})+i\nu\mathbb{T}(\mathbf{x})$ satisfies: $\operatorname{\operatorname{Im}}\left(\mathbb{M}_{\nu}(\mathbf{x})\mathbf{p}\cdot\overline{\mathbf{p}}\right)\geq \nu c_{\mathbb{T}}\|\mathbf{p}\|^2_{\mathbb{C}^

Figures (2)

  • Figure 1: An illustration to the simplified domain considered in Section \ref{['sec:simpl']}.
  • Figure 2: Illustration to the geometric configuration of Section \ref{['sec:domhole']}.

Theorems & Definitions (162)

  • Corollary 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Limiting absorption principle
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 152 more