Concentration of eigenfunctions near a concave boundary

TL;DR

This paper investigates how Laplacian eigenfunctions concentrate near a concave boundary on a 2D manifold, establishing inequalities that relate different bounds and proving one related to restrictions on broken geodesics.

Contribution

It introduces new inequalities connecting eigenfunction concentration bounds and proves a key inequality for restrictions to broken geodesics.

Findings

Bounded eigenfunction concentration near concave boundaries.

Established relationships between different eigenfunction inequalities.

Proved a new inequality for eigenfunction restrictions to broken geodesics.

Abstract

This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L^p norms of the restrictions of eigenfunctions to broken geodesics.