Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces
Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces
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DOI：
10.1016/j.jmaa.2024.128088
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发表时间：
2023-01
影响因子：
1.3
通讯作者：
T. Arias-Marco;E. Dryden;Carolyn S. Gordon;Asma Hassannezhad;Allie Ray;E. Stanhope
中科院分区：
文献类型：
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作者：
T. Arias-Marco;E. Dryden;Carolyn S. Gordon;Asma Hassannezhad;Allie Ray;E. Stanhope
我们考虑了与曲面上的Steklov问题和混合Steklov问题有关的三个不同问题。这些问题通过我们用来研究它们的技术联系在一起，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。在著名的关于Steklov特征值的Hersch-Payne-Schiffer和Weinstock不等式的精神下，考虑到混合Steklov-Neumann和Steklov-Dirichlet特征值的特征值之间的相互作用，我们得到了混合Steklov特征值的一个严格的等周不等式。1980年，Bandle证明了当k≤p−1时，单位圆盘使p阶单连通区域上的k个非零正规化Steklov本征值达到极大值.我们讨论了当k≥p时，单位圆盘是否仍是单连通旋转对称域类中的极大值.特别地，我们证明了当k→∞时，其上界收敛于Hersch-Payne-Schiffer上界.在假设Steklov边界与Dirichlet或Neumann边界交点的条件下，给出了任意曲面上混合Steklov问题的完全渐近性。
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 kth nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order p when k≤ p− 1. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when k≥ p. In particular, we show that as k→∞, 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.