Elliptische Probleme in der Allgemeinen Relativitätstheorie

The aim of Einstein's theory of gravity is to obtain a detailed understanding of all physically relevant space-times satisfying the Einstein equations. Mathematical studies of these equations belong to the domain of global geometric analysis. One is thus lead to many beautiful and challenging mathematical problems. The recent direct detection of gravitational waves makes it urgent to gain a deeper insight into several aspects of the theory.

One set of questions that arises is the rigorous construction of solutions of the Einstein equations with important physical features, as needed to justify the numerical simulations that are used to interpret the gravitational wave signals observed in the detectors.

Another issue is that of understanding of the nature and properties of black hole solutions. Indeed, the gravitational wave observed in September 2015 was emitted by a system of two black holes orbiting around each other, which collided to form a single one; this gave us the first proof of existence of nearby multiple black-hole systems in the universe. After a highly dynamical and extremely violent phase, where the energy equivalent of a total annihilation of three of our suns was emitted in a fraction of a second in form of gravitational waves, the final black hole settled down to a stationary one. A rigorous classification of such solutions is another key to a correct interpretation of the signals observed by the detectors.

The aim of this project is to address the above issues. This is done by using methods of the mathematical theory of “elliptic partial differential equations”. While the questions we address have been open for years, the new techniques recently introduced in the theory of elliptic equations will allow us to settle some of the key problems that remain, within a rigorous mathematical approach to the problems at hand.