What question did this study set out to answer?

The research aims to refine the upper bounds for Neumann eigenvalues in convex domains, particularly focusing on quantitative inequalities.

April 19, 2026Open Access

Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Key Points

The research aims to refine the upper bounds for Neumann eigenvalues in convex domains, particularly focusing on quantitative inequalities.
Deriving new inequalities for the k-th Neumann eigenvalue of a convex domain.
Using parameters such as diameter and the second largest semiaxis of the John ellipsoid.
Providing explicit values in the 2D planar case for specific constants.
Established new inequalities involving the diameter and specific constants for Neumann eigenvalues.
Demonstrated dependence of eigenvalue bounds on geometrical properties of convex domains.
Provided explicit bounds for the first eigenvalue in the plane.

Abstract

Refining the sharp upper bounds μ₊, ₃^* obtained by Kröger (1999) for the k-th Neumann eigenvalue of a convex domain Ω Rᵈ, we prove the following inequalities: for any k N there exists a constant C (k, d) >0 such that D_Ω² μₖ (Ω) μ₊, ₃^* - C (k, d) a₂ (Ω) ²/D_Ω² where D_Ω is the diameter of Ω and a₂ (Ω) is the second largest semiaxis of the John ellipsoid of Ω. In the planar case, for k=1 we also give an explicit value of the constant C (1, 2).

Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Key Points

Abstract

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