Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem

2023/04/22 by Xiong, Changwei · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2304.11297

Abstract

Let U⊂ ℝⁿ (n≥ 3) be an exterior Euclidean domain with smooth boundary ∂ U. We consider the Steklov eigenvalue problem on U. First we derive a sharp lower bound for the first eigenvalue in terms of the support function and the distance function to the origin of ∂ U. Second under various geometric conditions on ∂ U we obtain sharp upper bounds for the first eigenvalue. Along the proof, we get a sharp upper bound for the capacity of ∂ U when n=3 and ∂ U is connected. Last we also discuss an upper bound for the second eigenvalue.

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