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

Hans Christianson (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)

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

Published 12 August 2019