# Rellich-Christianson type identities for the Neumann data mass of Dirichlet eigenfunctions on polytopes

## TL;DR

This paper derives explicit Rellich-Christianson type identities linking Dirichlet eigenvalues to Neumann data on polytope faces, generalizing known results for simplices and providing formulas especially simple for polytopes with inscribed balls.

## Contribution

It introduces a new explicit formula connecting Dirichlet eigenvalues with Neumann data on polytope faces, extending previous identities from simplices to general polytopes.

## Key findings

- Explicit formula for Dirichlet eigenvalues in terms of Neumann data
- Simplified identities for polytopes with inscribed balls
- Generalization of Christianson's identities to polytopes

## Abstract

We consider the Dirichlet eigenvalue problem on a simple polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particular simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson [1][2].