STATIC SOLUTIONS TO THE SPHERICALLY SYMMETRIC EINSTEIN-VLASOV SYSTEM: A PARTICLE NUMBER-CASIMIR APPROACH

The existence of nontrivial static 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 nonlinear stability that gives this topic its importance. This problem was considered in [G. Wolansky, Arch. Ration. Mech. Anal., 156 (2001), pp. 205--230], but as has been pointed out in [H. Andre'\asson and M. Kunze, Arch. Ration. Mech. Anal., 235 (2020), pp. 783--791], that paper contained serious flaws. In this work we 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 Wolansky (2001). 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. Hence we settle rigorously parts of what Wolansky (2001) attempted to prove.