In the first part of this thesis, we discuss a new method for computing eigenvalues of elliptic operators defined on compact hypersurfaces. Standard methods for this problem are problematic, as they require either a parametrization of the hypersurface, which cannot always be found, or a good triangularization of the surface, which may be difficult to obtain. Our method, which avoids these requirements, computes the desired eigenvalues by solving a related elliptic eigenvalue problem in a cube containing the surface. We convert the original problem, defined on a hypersurface, to a problem defined on a cube using separation of variables and estimates from the semi-classical analysis of the Schrödinger operator. The latter problem is solved using standard finite element methods on the Cartesian grid. We also discuss the application of these ideas to solving evolution equations on surfaces, including a new proof of a result due to Greer (J. Sci. Comput. 29(3) 2006).

In the second part of this thesis, we consider the aggregation equation u_t + [special characters omitted] with nonnegative initial data in [special characters omitted] for n ≥ 2. We assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuation of solutions, with the bulk of the work devoted to proving existence. Because our PDE is a transport equation, existence for initial data in [special characters omitted] cannot be established using energy methods. Instead, we adapt ideas from the solution of the 2-D vorticity equation for the same class of initial data. Finite time blow-up (in the L^∞ norm) of solutions to our problem is proved when the kernel has precisely a Lipschitz point at the origin.