Generalised Rayleigh–Faber–Krahn inequality for polyharmonic eigenvalues

Let Ω\Omega be a domain of finite measure, let B\mathbb B denote the unit ball in Rd\mathbb R^d, and let ωd=B\omega_d=|\mathbb B|. For m,tNm,t\in\mathbb N, write λ1(m,t)(Ω)\lambda_{1}^{(m,t)}(\Omega) for the first eigenvalue of the buckling problem in the paper. Generalised Rayleigh–Faber–Krahn inequality. For any domain Ω\Omega of finite measure,

Ω2t/dλ1(m,t)(Ω)ωd2t/dλ1(m,t)(B).|\Omega|^{2t/d}\lambda_{1}^{(m,t)}(\Omega)\geq \omega_d^{2t/d}\lambda_{1}^{(m,t)}(\mathbb B).

This conjecture proposes that the ball minimises the volume-normalised first eigenvalue. Its status is not resolved in the supplied source.

Sources & referencesView supporting material

Primary source

Davide Buoso and Pedro Freitas, “Sharp inequalities and asymptotics for polyharmonic eigenvalues”, arXiv:2506.12791 (2025).

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