A one point non-concentration estimate for Laplace eigenfunctions on polygons

2018/08/09 by Hans Christianson, Christianson, Hans
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1808.03332

openalex publication_date 2018/08/09 · openalex created_date 2018/08/22 · openalex updated_date 2026/07/28

Abstract

In this paper we consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions. The main result is a quantitative estimate on the L² mass of eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face. In particular, eigenfunctions cannot concentrate completely at any one single point. The technique of proof is to use the commutator ideas from the recent work of the author \citeChr-tri,Chr-simp on triangles and simplices.

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