论文标题
爱因斯坦方程的初始边界价值问题，具有完全测量的时间表边界
The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary
论文作者
论文摘要
我们证明了具有唯一边界条件的爱因斯坦方程的初始边界值问题的良好性，要求及时边界完全测量。这为这种特定的几何边界条件和第一个设置提供了第一个良好的结果，而弗里德里希原始意义上的几何唯一性则为初始边界值问题所具有的第一个设置。我们的证明依赖于爱因斯坦真空方程的ADM系统，该方程是根据沿及时的测量学沿平行的繁殖正顺序框架提出的。作为一个独立的结果，我们首先在爱因斯坦方程的凯奇问题的量规中建立了良好的性，包括约束的传播。更确切地说，我们表明，通过适当修改进化方程，使用约束方程，我们可以为正统帧的连接系数得出一阶对称双曲系统。然后，约束的传播依赖于涉及连接，适当修改的Riemann和Ricci曲率张量的双曲线系统的推导以及连接的扭转。特别是，该连接显示与约束的有效性同时与Levi-Civita连接一致。在最初的边界值问题的情况下，我们随后验证了边界的第二个基本形式消失会导致我们修改后的ADM系统以及用于传播约束的双曲线系统的均质边界条件。额外的分析难度是由于无法控制溶液边界的正常衍生物的丧失。为了解决此问题，我们使用各向异性的Sobolev空间规模，并利用方程式的特定结构。
We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent result, we first establish the well-posedness in this gauge of the Cauchy problem for the Einstein equations, including the propagation of constraints. More precisely, we show that by appropriately modifying the evolution equations, using the constraint equations, we can derive a first order symmetric hyperbolic system for the connection coefficients of the orthonormal frame. The propagation of the constraints then relies on the derivation of a hyperbolic system involving the connection, suitably modified Riemann and Ricci curvature tensors and the torsion of the connection. In particular, the connection is shown to agree with the Levi-Civita connection at the same time as the validity of the constraints. In the case of the initial boundary value problem with totally geodesic boundary, we then verify that the vanishing of the second fundamental form of the boundary leads to homogeneous boundary conditions for our modified ADM system, as well as for the hyperbolic system used in the propagation of the constraints. An additional analytical difficulty arises from a loss of control on the normal derivatives to the boundary of the solution. To resolve this issue, we work with an anisotropic scale of Sobolev spaces and exploit the specific structure of the equations.
