Upper bound for the first p-Steklov eigenvalue in ℝⁿ

2026/07/23 by Lili Wang, Tao Wang
#math.AP #math.DG #math.SP

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Abstract

For p∈(1,n] and any bounded convex domain Ω⊂ℝⁿ, we prove the sharp inequality Λ_p(Ω):=\fracW_p(Ω)P(Ω)V(Ω)^p/n≥ω_n^-p/n, W_p(Ω)=∫_∂Ω|x|^p dS, with equality holding exactly at centered balls. Combining this with the isoperimetric inequality yields the explicit upper bound σ_1,p(Ω)≤ \fracA(n,p)r(Ω^*)^p-1, where Ω^* is a ball having the same perimeter as Ω, and A(n,p)=1 for 1<p≤ 2, A(n,p)=n^p/2-1 for 2<p≤ n. When p=2, the result recovers the higher-dimensional Weinstock inequality of Bucur et al. [J. Differential Geom. 2021]. We also obtain an explicit upper bound for the first Wentzell eigenvalue of the p-Laplacian on convex domains.

Related

• Weinstock inequality in higher dimensions
• A quantitative Weinstock inequality
• An upper bound for the first nonzero Steklov eigenvalue
• Upper bounds for the Steklov eigenvalues of the p-Laplacian
• The exterior Steklov problem for Euclidean domains