具有自由边界的极小超曲面的第一个四阶Steklov特征值的估计

Pub Date : 2023-09-03 DOI:10.2140/pjm.2023.325.1

Rondinelle Batista, Barnab'e Lima, Paulo Sousa, Bruno Vieira

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引用次数: 0

摘要

研究了具有非负Ricci曲率和严格凸边界的（n+1）维紧致流形Mn+1中具有自由边界的紧致嵌入超曲面（cid:54）n的四阶Steklov问题。如果（cid:54）是极小的，我们建立了这个问题的第一特征值的下界。当M=Bn+1是（cid:82）n+1中的单位球时，如果（cid:54）具有恒定的平均曲率H（cid:54），我们证明了第一特征值满足σ1≤n+|H（cid:54）|。在极小情况（H（cid:54）=0）下，我们证明了σ1=n。

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Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary

. We explore the fourth-order Steklov problem of a compact embedded hyper-surface (cid:54) n with free boundary in a ( n + 1 ) -dimensional compact manifold M n + 1 which has nonnegative Ricci curvature and strictly convex boundary. If (cid:54) is minimal we establish a lower bound for the first eigenvalue of this problem. When M = B n + 1 is the unit ball in (cid:82) n + 1 , if (cid:54) has constant mean curvature H (cid:54) we prove that the first eigenvalue satisfies σ 1 ≤ n + | H (cid:54) | . In the minimal case ( H (cid:54) = 0), we prove that σ 1 = n .

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