# Equidistribution of Neumann data mass on triangles

## TL;DR

This paper proves an exact formula for the distribution of Neumann data of Dirichlet eigenfunctions on triangles, showing the $L^2$ norm on each side relates precisely to the side length and triangle area.

## Contribution

It provides the first exact, non-asymptotic formula for Neumann data distribution on triangle boundaries, using simple integration techniques.

## Key findings

- The $L^2$ norm of Neumann data on each side equals side length divided by triangle area.
- The result is an exact formula, not asymptotic.
- The proof uses straightforward integration by parts.

## Abstract

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on triangles. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each side is equal to the length of the side divided by the area of the triangle. The novel feature of this result is that it is {\it not} an asymptotic, but an exact formula. The proof is by simple integrations by parts.