On the Dirichlet problem for the stationary and axisymmetric Einstein equations (Q1270314)

In this interesting paper, the author studies Einstein's field equations for a rigidly rotating perfect fluid in equilibrium. In suitable coordinates these equations can be written as a semilinear system of six purely elliptic PDEs of second-order under Dirichlet boundary condition. Due to the high nonlinearity of these field equations no general existence or uniqueness theorems are known hitherto for such problems. The author shows that the Dirichlet problem for the vacuum region outside a ball, and for a ball inside the matter region, has a unique regular solution if the boundary data are in a characteristic way limited by the ``diameter'' of the ball. Moreover, the used methods are directly related to a numerical solution technique for axisymmetric rotating relativistic bodies due to Bonazzola et al. (1993).