On hearing the shape of rectilinear regions

Abstract

From complete knowledge of the eigenvalues of the negative Laplacian on a bounded domain, one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function. Explicit calculations are performed for an equilateral triangular domain with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, some applications to eigenvalue problems for an equilateral triangular prism are dealt with, including the solid vertex terms.