Eigenvalues for the Laplace Operator in the Interior of an Equilateral Triangle

The eigenvalues of the Laplace operator for the Dirichlet, Neumann and Robin problems in the interior of an equilateral triangle were first obtained by Lamé. Here, we present a simple, unified approach for rederiving the above results and also obtain the eigenvalues for the oblique Robin and for certain Poincaré problems. The explicit formula for the Poincaré eigenvalues yields, via appropriate limits, the relevant formulae for the oblique Robin, Robin, Neumann and Dirichlet eigenvalues. The method introduced here is based on the analysis of the so-called global relation, which as shown recently in the literature, provides an effective tool for the study of boundary value problems.