Max Planck Institute for Gravitational Physics

This is a joint work with Pin Yu. For semi-linear wave equations with null form non-linearities on $\mathbb{R}^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.

In this talk I will discuss the Cauchy problem for bounded

self-gravitating elastic bodies in Einstein gravity. One of the main

difficulties is caused by the fact that the spacetime curvature must be

discontinuous at the boundary of the body. In order to treat the Cauchy

problem, one must show that the jump in the curvature propagates along

the timelike boundary of the spacetime track of the body. I will discuss

a proof of local well-posedness which takes this behavior into account.

Self-gravitating elastic bodies provide models for extended

objects in general relativity. I will discuss constructions of static

and rotating self-gravitating bodies, as well as recent results on the

initial value problem for self-gravitating elastic bodies.