Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability

Abstract

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality for the principal frequency of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995]. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N is close enough to 2 or 3. The inequality contains the so-called ``asymptotic volume ratio", and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the RCD(0,N) space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied.

arXiv comment30 pages

References in corpus (12)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications
• Topological regularity of isoperimetric sets in PI spaces having a deformation property
• Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate
• Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate
• Failure of strong unique continuation for harmonic functions on RCD Spaces