This project is to study the first nonzero eigenvalue and the corresponding eigenfunction for the Neumann Laplacian on convex domains in three- dimensional space and also on two- dimensional convex manifolds. Particular attention will be devoted to describing the location of the nodal set and the maxima of the eigenfunction. This will be done by reducing the problem to the analysis of an eigenproblem for an ordinary differential operator whose coefficients are determined by the geometry of the domain. The methods are based on the use of variational methods, energy integrals, maximum principles and gradient estimates.
The results of this research should provide proofs of results that are of importance in acoustics, elasticity, thermodynamics and electrodynamics. The description of solutions of both the heat equation and the wave equation often are governed by expressions in which the first eigenfunction plays a special role.
