UC Berkeley

This dissertation deals with the question of well-posedness for the three-form field equation in eleven dimensional supergravity and the minimal time-like hypersurface equation in Minkowski space. Both of these equations are nonlinear wave equations.

The three-form field equation arises in the classical field theory formulation that underlies string theory. After making simplifying assumptions, it becomes a system for a differential three-form on $\R^{3+1}\times K^7$, where $K$ is a compact Riemannian manifold. A suitable gauge choice turns the the equation into a semilinear wave equation. We show that the three-form field equation is globally in time well posed for small, smooth, compactly supported initial data.

The minimal surface equation in Minkowski space $\R^{(n+1)+1}$ describes a submanifold, which has a vanishing mean curvature. If the submanifold is time-like, the equation is a quasilinear wave equation. We prove that for $n=2,3$ and under some further geometrical assumptions, the equation is locally well posed for initial data in $H^\frac{n+3}{2}(\R^n)\times H^\frac{n+1}{2}(\R^n)$.

The nonlinearities in the equations above belong to a restricted class of nonlinearities that are said to satisfy the null condition. The goal of the null condition is to select the nonlinearities that limit the amount of interactions between waves that have a small angle of incidence. The precise formulation of the null condition depends on the context of the Cauchy problem and it is not yet clear in all cases. In particular, our investigation of the minimal surface equation falls in the context of questions regarding large data quasilinear equations for which there is still only a conjectured formulation of the condition. We hope that our treatment of the particular example can shed some light on the general case.