Capacity estimates and improved lower bounds for the inner radius of nodal domains

2026/08/02 by Philippe Charron
Mathematics · #math.SP #math.AP

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arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We show that for every closed, smooth manifold (M,g) of dimension d, there exists c(g) such that any nodal domain Ω_λ of a Laplace eigenfunction with eigenvalue λ contains a geodesic ball of radius at least c(g) λ^-1/2 loglog(λ)^-1/2 if d=3 and c(g) λ^-1/2 log(λ)^-(d-3)/(2) if d >3. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any d ≥ 3, there exist sequences of λ-nodal domains on \mathbbT^d whose inner radius is of order o(λ^-1/2).

Citations

• The inner radius of nodal domains in high dimensions
• Localization properties of high energy eigenfunctions on flat tori
• High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology
• Quantitative propagation of smallness for solutions of elliptic equations
• On the lower bound of the inner radius of nodal domains
• Bounded solutions to the Allen-Cahn equation with level sets of any\n compact topology