Counting the Eigenvalues of the Laplace Operator on Some Convex Domains

We define algorithms for computing the eigenvalue counting functions (ECFs) of the Laplacian with Dirichlet and Neumann boundary conditions for a variety of convex 2D and 3D domains using floating point precision in Python.These domains include the square, disk, ball, cube, and cylinder.We use these computed ECFs to validate Polya's conjecture on these domains.The conjecture has not been proven in completeness on the ball, so this work presents numerical evidence to the positive.We also use these calculations to study the potential geometric information in the lower-order terms of the asymptotics by comparing ECFs between different shapes.We do this by subtracting the like terms in the asymptotics of the ECFs to draw out the lower-order information.In the 3D case, a difference can be seen in the lower-order asymptotics of a cube and cylinder with the same volume and surface area.In particular, the cube's o(k 2 ) term has a larger variance than the cylinder's term, and in both cases, the term is mostly positive.