Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifolds

2024/06/23 by Kévin Le Balc’h, Balc'h, Kévin Le
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2406.16036

Abstract

We study local growth properties of Laplace eigenfunctions on compact Riemannian manifolds. Following the paradigm introduced by Donnelly and Fefferman in the late 1980s, an eigenfunction is expected to behave locally like a polynomial of degree comparable to the square root of the eigenvalue. In this direction we establish almost sharp local L^p--Bernstein inequalities, p∈[1,∞], conjectured by Donnelly--Fefferman in 1990. We also derive analogous estimates for A-harmonic functions, with the square root of the eigenvalue replaced by the doubling index. Our argument refines the original Donnelly--Fefferman method based on L²--Carleman estimates. At the L²--level, we first prove a uniform bound for the doubling index on annuli of width comparable to the wavelength. This implies, with an arbitrarily small polynomial loss, the corresponding property at the L^p--level for all p∈[1,∞]. The latter step relies on a bootstrap scheme combining elliptic regularity with a patching of local Carleman estimates on small balls.

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