<p>The paper studies the properties of acoustic operators in bounded Lipschitz domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with m-dissipative generalized impedance boundary conditions, which are used to model open acoustic and optical cavities. We prove that such acoustic operators have compact resolvent if and only if the impedance operator from the trace space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^{1/2} (\partial \Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to the other trace space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^{-1/2} (\partial \Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is partially based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations and to models involving conservative or lossy resonators. In order to study impedance boundary conditions with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-coefficients, we consider the classes of pointwise multipliers between fractional Sobolev spaces on Lipschitz boundaries <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. Examples of m-dissipative boundary conditions that lead to acoustic operators with non-empty essential spectra are also provided.</p>

M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces

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2026年4月4日

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Illya M. Karabash

摘要

The paper studies the properties of acoustic operators in bounded Lipschitz domains $\Omega $ $Ω$ with m-dissipative generalized impedance boundary conditions, which are used to model open acoustic and optical cavities. We prove that such acoustic operators have compact resolvent if and only if the impedance operator from the trace space $H^{1/2} (\partial \Omega )$ $H^{1 / 2} (\partial Ω)$ to the other trace space $H^{-1/2} (\partial \Omega )$ $H^{- 1 / 2} (\partial Ω)$ is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is partially based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations and to models involving conservative or lossy resonators. In order to study impedance boundary conditions with $L^q$ $L^{q}$ -coefficients, we consider the classes of pointwise multipliers between fractional Sobolev spaces on Lipschitz boundaries $\partial \Omega $ $\partial Ω$ . Examples of m-dissipative boundary conditions that lead to acoustic operators with non-empty essential spectra are also provided.