Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known.
However, it is an open problem whether or not static solutions arise as minimizers of a variational problem.
Apart from being of interest in its own right, it is the connection to non-linear stability that gives this
topic its importance. This problem was considered in [1], but as has been pointed out in [2],
the paper [1] contained serious flaws. In this talk we will show how to construct static solutions by solving
the Euler-Lagrange equation for the energy density $\rho$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in [1]. We then define a density function $f$ on phase space which induces the energy density $\rho$ and we show that it constitutes a static solution of the Einstein-Vlasov system.
This is a joint work with Hakan Andreasson.
[1] Wolansky G.: Static solutions of the Vlasov-Einstein system,
Arch. Ration. Mech. Anal. 156, 205-230 (2001)
[2] Andreasson H. & Kunze M.: Comments on the paper `Static solutions of the Vlasov-Einstein system' by G. Wolansky, Arch. Ration. Mech. Anal. 235, 783-791 (2020)