On the Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

Abstract:

In suitable coordinates Einstein's field equations for a rigidly rotating perfect fluid in equilibrium can be written as a semilinear system of purely elliptic partial differential equations of second order. Therefore, the formulation of a boundary value problem is appropriate in this situation. It is shown 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. This restriction seems to be closely connected with stability limits for rotating stars. Furthermore, the used mathematical methods are directly related to a numerical solution technique for such physical systems.