Mathematical Research Letters
Volume 26 (2019)
Equidistribution of Neumann data mass on simplices and a simple inverse problem
Pages: 421 – 445
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