<p>We study the asymptotic behavior of the electric field in the transverse magnetic (TM) mode, propagating in a domain consisting of a two-dimensional ball surrounded by a thin ring of thickness <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1230_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> (detined to tend to 0), with contrast coefficients tending to infinity, and embedded in an ambient medium. We derive and justify an asymptotic expansion of the solution <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1230_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$u_{\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> to the Helmholtz problem with respect to the thickness <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1230_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>. Subsequently, we establish Ventcel transmission conditions on the limiting interface G modeling the effect of the thin layer with accuracy up to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1230_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$O(\varepsilon ^{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

On Ventcel transmission conditions for a Helmholtz problem with a high contrast thin ring

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发表日期：

2025年4月30日

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Khaled El-Ghaouti Boutarene

摘要

We study the asymptotic behavior of the electric field in the transverse magnetic (TM) mode, propagating in a domain consisting of a two-dimensional ball surrounded by a thin ring of thickness $\varepsilon $ $ε$ (detined to tend to 0), with contrast coefficients tending to infinity, and embedded in an ambient medium. We derive and justify an asymptotic expansion of the solution $u_{\varepsilon }$ $u_{ε}$ to the Helmholtz problem with respect to the thickness $\varepsilon $ $ε$ . Subsequently, we establish Ventcel transmission conditions on the limiting interface G modeling the effect of the thin layer with accuracy up to $O(\varepsilon ^{3})$ $O (ε^{3})$ .