Near-boundary rigidity and strict maximum modulus for Neumann eigenfunctions on convex planar domains

Abstract

We prove that the maximum modulus of every nonzero second Neumann eigenfunction on a bounded convex planar domain $Ω$ is attained only on the boundary, without boundary smoothness or eigenvalue simplicity. The key is a rigidity theorem for any positive Neumann eigenvalue $μ$: an interior maximum-modulus point $p$ with $\sqrtμ\,\operatorname{dist}(p,\partialΩ)<π/\sqrt{3}$ forces the domain to be a rectangle and the eigenfunction to be a cosine of one coordinate. In particular, $μ\ge4μ_2(Ω)$, and the factor $4$ is sharp. This result reveals how a local distance condition at a single interior maximum-modulus point can enforce global geometric rigidity, thereby linking extremum localization to the classification of domains and eigenfunctions.