For every given <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta <0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we study the problem of maximizing the first Robin eigenvalue of the Laplacian <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _\beta (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>β</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> among convex (not necessarily smooth) sets <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset {\mathbb {S}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with fixed perimeter. In particular, denoting by <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> the perimeter of the n-dimensional hemisphere, we show that for fixed perimeters <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$P<\sigma _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo><</mo> <msub> <mi>σ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2007_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and the ball D of the same perimeter.

A Spectral Isoperimetric Inequality on the n-Sphere for the Robin-Laplacian with Negative Boundary Parameter

期刊论文
出版模式：

开放获取
发表日期：

2025年4月28日

查看全文

P. Acampora,
A. Celentano,
E. Cristoforoni,
C. Nitsch,
C. Trombetti

摘要

For every given $\beta <0$ $β < 0$ , we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda _\beta (\Omega )$ $λ_{β} (Ω)$ among convex (not necessarily smooth) sets $\Omega \subset {\mathbb {S}}^{n}$ $Ω \subset S^{n}$ with fixed perimeter. In particular, denoting by $\sigma _n$ $σ_{n}$ the perimeter of the n-dimensional hemisphere, we show that for fixed perimeters $P<\sigma _n$ $P < σ_{n}$ , geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between $\Omega $ $Ω$ and the ball D of the same perimeter.