In this lecture, we discuss some of our recent work on the number of solutions of weakly nonlinear elliptic partial differential equations. In particular, we discuss cases where the answers were surprising (at least to me). In $\S 1$, we have rather fewer solutions than one might expect while in $\S 2$, we have rather more solutions than one might expect. The discussion in $\S 1$ is rather brief since a more complete discussion appears in [4].

In this paper I wish to discuss the classical solvability of the first boundary value problem for a class of non-linear parabolic equations of second order. The equations to be considered arise from symmetric functions in a natural way analagous to the equations considered by Caffarelli, Nirenberg and Spruck [CNS] in the elliptic case. They are also motivated by the proposed analogue of the Monge-Ampère equation of Krylov [KL which is considered here as a first special case. I do not present the proofs of the results described, but only rough indications of the methods involved. The work constitutes the central results of the latter half of my doctoral dissertation [R l]