可能隐对称性在曲面Steklov和混合Steklov问题中的应用
Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces
摘要：
我们考虑三个不同的问题有关的Steklov和混合Steklov问题的表面。这些问题与我们用来研究它们的技术有关，这些技术以各种方式利用对称性，即使我们研究的表面不一定具有固有的对称性。在著名的Steklov特征值的Hersch-Payne-Schiffer不等式和Weinstock不等式的基础上，考虑混合Steklov-Neumann特征值和Steklov-Dirichlet特征值之间的相互影响，得到了混合Steklov特征值的一个尖锐的等周不等式. 1980年，Bandle证明了当$k\le p-1$时，单连通区域上具有$p$阶旋转对称性的单位圆盘极大化了$k$阶非零归一化Steklov特征值.本文讨论了当$k\geq p$满足条件时，圆盘在单连通旋转对称整环类中是否保持最大化。特别地，我们证明了对于足够大的$k$，上界收敛到Hersch-Payne-Schiffer上界。我们给出了任意曲面上的混合Steklov问题的完全渐近性，假设在Steklov边界与Dirichlet边界或Neumann边界的交点处存在一些条件。
英文摘要：
We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the $k$th nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order $p$ when $k\le p-1$. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when $k\geq p$. In particular, we show that for $k$ large enough, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
