David Fajman

Nodal domains of a non-separable problem—the right-angled isosceles triangle

We study the nodal set of eigenfunctions of the Laplace operator on the right-angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number νn of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually, we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes. Furthermore, we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics.

Local Well-Posedness for the Einstein--Vlasov System

We prove a local well-posedness result for the Einstein--Vlasov system in constant mean curvature--spatial harmonic gauge introduced in [L. Andersson and V. Moncrief, Ann. Henri Poincare, 4 (2003), pp. 1--34], where local well-posedness for the vacuum Einstein equations is established. This work is based on the techniques developed therein. In addition, we use the regularity theory and techniques for proving the existence of solutions to the Einstein--Vlasov system, recently established in [H. Ringstrom, Oxford Math. Monogr., 2013], where the local stability problem for the Einstein--Vlasov system is solved in generalized harmonic gauge.

Future asymptotic behavior of three-dimensional spacetimes with massive particles

We consider non-vacuum initial data for the three-dimensional Einstein equations coupled to Vlasov matter composed of massive particles, on an arbitrary compact Cauchy hypersurface without boundary. We show that conservation of the total mass implies future completeness of the corresponding maximal development in the isotropic case, independent of the topology. This behavior is fundamentally different from the vacuum case and also from the same model in higher dimensions. In particular, we find that a positive mass of particles in three dimensions avoids recollapse of the spatial geometry. Finally, we construct similar solutions for the Einstein-dust system and describe to what extent the construction fails for massless matter models.

The Einstein−Λ flow on product manifolds

We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form . In dimensions we show the existence of continuous families of recollapsing models whenever at least one of the factors M 1 or M 2 admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which exhibit fundamentally different types of long-time behavior under the Einstein– flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related to the spatial topology through the existence of Riemannian Einstein metrics of positive curvature.

Static Solutions to the Einstein--Vlasov System with a Nonvanishing Cosmological Constant

We construct spherically symmetric static solutions to the Einstein--Vlasov system with nonvanishing cosmological constant

The nonvacuum Einstein flow on surfaces of nonnegative curvature

\Lambda

Courant-sharp eigenvalues of Neumann 2-rep-tiles

. The results are divided as follows. For small

On the CMC-Einstein-Λ flow

\Lambda>0

A vector field method for relativistic transport equations with applications

we show the existence of globally regular solutions which coincide with the Schwarzschild--deSitter solution in the exterior of the matter regions. For

Sharp asymptotics for small data solutions of the Vlasov-Nordstr\"om system in three dimensions

\Lambda<0