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Eigenstructure of the equilateral triangle

Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored. 1. Introduction. The eigenstructure of the Laplacian on an equilateral triangle under either Dirichlet or Neumann boundary conditions was explicitly determined by Lamé [6, 7] in the context of his studies of heat transfer in polyhedral bodies and then further explored by Pockels [12]. However, Lamé and subsequent researchers such as Pockels did not provide a complete derivation of these formulas but rather simply stated them and then proceeded to show that they satisfied the relevant equation and associated boundary conditions. Such a complete, direct, and elementary derivation of Lamé's formulas has only recently been provided for the Dirichlet problem [11] as well as the Neumann problem [9]. It is the express purpose of the present work to extend this recent work to the much more difficult case of the Robin boundary condition. Lamé [6, 7] presented a partial treatment of this problem when he considered eigenfunctions possessing 120 • rota