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: Let $f(t)$ be a convex, positive, increasing nonlinearity. It is known that stable solutions of $-\Delta u =f(u)$ can be singular (i.e., unbounded) if the dimension $n \ge 10$.

Brezis conjectured that if $x=0$ is such a singular point, then $f'(u(x))$ blows-up like $|x|^{2-n}$. Villegas showed that such a strong statement fails for general nonlinearities.

In this talk, we prove — for all nonlinearities — a version of Brezis conjecture, which is essentially the best one can obtain in view of the counterexamples of Villegas. Building on this result we then show that the singular set has dimension n-10, at least for a large class of nonlinearities that includes the most relevant cases. This is a joint work with Alessio Figalli.

Abstract: A central problem in general relativity concerns the formation of naked singularities, a class of finite-time blowup solutions to Einstein-matter systems with starkly different properties than their black hole counterparts. In this talk we introduce the family of k-self-similar naked singularities, first constructed rigorously by Christodoulou, which are known to exhibit a blue-shift instability. In our main results, we quantify the strength of this blue-shift instability as the degree of concentration and the support of initial data are varied, identifying surprising regimes in which these spacetimes transition between stability and instability. We then discuss the consequences of these results for the weak cosmic censorship conjecture.

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.