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# Mathematical Research Letters

## Volume 26 (2019)

### Number 2

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

Pages: 421 – 445

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n2.a4

#### Author

#### 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 “Equidistribution of Neumann data mass on triangles” [H. Christianson, *Proc. Amer. Math. Soc.* 145 (2017), no. 12, 5247–5255] to higher dimensions. Again it is *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 *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.

The author’s work was supported in part by NSF grant DMS-1500812.

Received 7 August 2017

Accepted 12 June 2018

Published 12 August 2019