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arXiv · 2208.10541

On a Bernstein inequality for eigenfunctions

Stefano Decio·Eugenia Malinnikova
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Abstract

Let $φ_λ$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $Δ_g φ_λ + λφ_λ=0$. We show that $φ_λ$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nablaφ_λ|\leq C_δ\max\left\{\frac{\sqrtλ\log^{2+δ}λ}{r},λ\log^{2+δ}λ\right\}\sup_{B_g(x,r)}|φ_λ|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.

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Stefano Decio, Eugenia Malinnikova. 2023-01-31. On a Bernstein inequality for eigenfunctions. https://arxiv.org/abs/2208.10541

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