MIT PDE/Analysis Seminar

Abstract: Given a subset of Euclidean space, one may consider the space of harmonic functions vanishing on it. In two dimensions, this space is always either trivial or infinite-dimensional. In higher dimensions, this is no longer true. In this talk we will see that almost all cones defined by a quadratic homogeneous harmonic polynomial admit exactly two linearly independent harmonic functions vanishing on them.

This phenomenon also generalizes in a natural way to solutions of second order elliptic PDEs with smooth coefficients.

Abstract: In this talk, we will focus on the local growth properties of Laplace eigenfunctions on a compact Riemannian manifold. The principal theme is that a Laplace eigenfunction behaves locally as a polynomial function of degree proportional to the square root of the eigenvalue. More precisely, we will discuss local Bernstein estimates for Laplace eigenfunctions, conjectured a while ago by Donnelly and Fefferman.