586 ON SHAPE OPTIMIZING THE RATIO OF THE FIRST TWO EIGENVALUES OF THE LAPLACIAN J. Haeberly, October 1991 We investigate numerically a 1956 conjecture of Payne, Polya, and Weinberger. The conjecture asserts that the ratio of the first two eigenvalues of the Laplacian on a bounded domain \Omega of the plane with Dirichlet boundary conditions reaches its minimum value precisely when \Omega is a disk. A crucial feature of this problem is the loss of smoothness of the objective function at the solution. The following results form the core of our numerical treatment. First, we construct finite dimensional families of deformations of a disk equipped with a uniform triangulation. This permits the formulation of a discrete model of the problem via finite element techniques. Second, we build on the work of M. Overton to derive optimality conditions in terms of Clarke's generalized gradients for nonsmooth functions. These ideas are then combined into an algorithm and implemented in Fortran.