TY - JOUR
T1 - A Rigidity Result for the Schiffer Conjecture on Domain with a Hole
AU - Sun , Yingxin
JO - Analysis in Theory and Applications
VL - 3
SP - 229
EP - 237
PY - 2025
DA - 2025/09
SN - 41
DO - http://doi.org/10.4208/ata.OA-2024-0023
UR - https://global-sci.org/intro/article_detail/ata/24484.html
KW - Schiffer conjecture, overdetermined problem, symmetry.
AB - <p style="text-align: justify;">Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution
 to the problem&nbsp;<br/></p><p style="text-align:center"><img src="https://admin.global-sci.org/uploads/ueditor/image/20250929/1759127283706666.png" title="1759127283706666.png" alt="2.png" width="388" height="42"/></p><p style="text-align: justify;"><span style="text-align: justify;">where $\partial^{\pm}\Omega$ represents the outer and inner boundaries of $\Omega,$ respectively, $c$ is a constant.
 Let ${\mu}_k$ denote the $k{\rm th}$ Neumann eigenvalue of the Laplacian on $\Omega$ and${\Omega}_h$ is the hole.
 We establish that if $\mu&lt; {\mu}_8,$ then $\Omega$ is an annulus.</span><br/></p>