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

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.

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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