AI 中文总结
该研究将弯曲空间的标度结果从拉普拉斯算子扩展到四阶特征值问题，推导了球面和双曲空间中测地圆盘上夹紧板特征值的标度不等式，明确了特征值的极限行为，并通过共形反演关联了球面屈曲极限的负发散与外部屈曲问题。
AI 中文摘要
对于球面和双曲空间中的双拉普拉斯算子，测地圆盘上的夹紧振动特征值与屈曲特征值被证明满足标度不等式，该不等式与欧几里得板特征值的标准标度不变性类似。这些结果将弯曲空间的标度结果从拉普拉斯算子扩展到四阶特征值问题。此外，还确定了当测地圆盘膨胀至充满周围空间时，标度特征值的极限行为：根据几何形状和标度选择的不同，特征值会趋于零、正无穷、负无穷，或与双曲谱底部相关的有限值。其中一个球面屈曲极限中的负发散，通过共形反演与外部屈曲问题相关联。
英文摘要
For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.
