Abstract
We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by K>0 and dimension bounded above by N∈(1,∞) in a synthetic sense, the so called CD(K,N) spaces. We first establish a Polya-Szego type inequality stating that the W1,p-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p-Laplace operator with Dirichlet boundary conditions (on open subsets), for every p∈(1,∞). This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41]; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci ≥K>0, finite dimensional Alexandrov spaces with curvature≥K>0, Finsler manifolds with Ricci ≥K>0.In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of RCD(K,N) spaces, which are interesting even for smooth Riemannian manifolds with Ricci ≥K>0.
Highlights
In 1884 Lord Rayleigh, in his book about the theory of sound [46], conjectured that, among all membranes of a given area, the disk has the lowest fundamental frequency of vibration
We study decreasing rearrangements of functions defined on metric measure spaces with Ricci curvature bounded below by K > 0 and dimension bounded above by N ∈ (1, ∞) in a synthetic sense, the so called CD(K, N ) spaces
The first Dirichlet eigenvalue of Ω is bounded below by the first Dirichlet eigenvalue of a Euclidean ball having the same volume of Ω, the inequality is rigid in the sense that equality is attained if and only if Ω is a ball
Summary
In 1884 Lord Rayleigh, in his book about the theory of sound [46], conjectured that, among all membranes of a given area, the disk has the lowest fundamental frequency of vibration. In the same spirit as above, for a function u ∈ Cc1(M ) define a spherical decreasing rearrangement u∗ on S; second, replace the Euclidean isoperimetric inequality by the Lévy-Gromov isoperimetric inequality [30, Appendix C] in the proof of the corresponding Polya-Szego. In order to state the main theorems, let us introduce some notation about the model one-dimensional space and the corresponding monotone rearrangement. Let us stress that the condition |∇u| = 0 m-a.e. is necessary to infer that u(·) = u∗ ◦ d(x0, ·), even knowing a priori that the space is a spherical suspension with pole x0 and that u achieves equality in Polya-Szego inequality.
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