半正多边形的谱确定
Spectral determination of semi-regular polygons
摘要：
我们说一个有$n$边的多边形是半正则的，如果它是可外切的，并且它的角都相等，但可能只有一个，这个角大于其余的角。特别地，正多边形是半规则的。证明了半正多边形在凸分片光滑域类中是谱决定的。具体地说，我们证明了如果$Ω$是一个凸的分段光滑平面区域，可能有直的角，其Dirichlet或Neumann谱与$n$边的半正多边形$P_n$的Dirichlet或Neumann谱重合，则$Ω$与$P_n$全等.
英文摘要：
Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $Ω$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $Ω$ is congruent to $P_n$.
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