# Equidistribution of Neumann data mass on simplices and a simple inverse problem

## TL;DR

This paper establishes an exact formula for the distribution of Neumann data of Dirichlet eigenfunctions on simplices across dimensions and investigates whether these data norms determine elliptic operators, with results varying by dimension.

## Contribution

It generalizes previous 2D results to higher dimensions with an exact formula and explores the inverse problem of determining elliptic operators from Neumann data norms.

## Key findings

- Neumann data norms are proportional to face volumes in all dimensions.
- Exact formula for Neumann data distribution on simplices, not asymptotic.
- Inverse problem: data norms determine elliptic operators in 2D, but not in higher dimensions.

## Abstract

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on simplices. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each face is equal to $2/n$ times the $(n-1)$-dimensional volume of the face divided by the volume of the simplex. This is a generalization of \cite{Chr-tri} to higher dimensions. Again it is {\it not} an asymptotic, but an exact formula. The proof is by simple integrations by parts and linear algebra. We also consider the following inverse problem: do the {\it norms} of the Neumann data on a simplex determine a constant coefficient elliptic operator? The answer is yes in dimension 2 and no in higher dimensions.