紧致流形上的Lichnerowicz方程
The Lichnerowicz equation on compact manifolds with boundary
摘要：
本文系统地研究了紧致流形上的Einstein约束方程的适定性问题。这是广义相对论中的一个重要问题，在数值相对论中尤其重要，因为它出现在包含渐近平坦端和/或捕获面的柯西曲面模型中。此外，在发展开放的渐近欧氏流形的解理论时出现的一些技术障碍在有边界的紧致流形上有类似之处。作为第一步，在这里，我们限制自己的Lichnerowicz方程，也称为哈密顿约束方程，这是在约束系统的非线性的主要来源。重点是低规律性数据和不同类型的边界条件之间的相互作用，这是以前没有仔细分析。为了发展一个适定性理论，反映了现有的理论的情况下，闭流形，我们首先推广Yamabe分类的非光滑度量的紧流形上的边界。然后我们将共形不变性的结果推广到有边界的流形，并证明了一个唯一性定理。最后，通过使用子解和上解的方法（保序映射迭代），我们建立了几个存在性结果的一大类问题，涵盖了广泛的参数制度，其中包括大多数的情况下，在实践中相关的。
英文摘要：
In this article we initiate a systematic study of the well-posedness theory of the Einstein constraint equations on compact manifolds with boundary. This is an important problem in general relativity, and it is particularly important in numerical relativity, as it arises in models of Cauchy surfaces containing asymptotically flat ends and/or trapped surfaces. Moreover, a number of technical obstacles that appear when developing the solution theory for open, asymptotically Euclidean manifolds have analogues on compact manifolds with boundary. As a first step, here we restrict ourselves to the Lichnerowicz equation, also called the Hamiltonian constraint equation, which is the main source of nonlinearity in the constraint system. The focus is on low regularity data and on the interaction between different types of boundary conditions, which has not been carefully analyzed before. In order to develop a well-posedness theory that mirrors the existing theory for the case of closed manifolds, we first generalize the Yamabe classification to nonsmooth metrics on compact manifolds with boundary. We then extend a result on conformal invariance to manifolds with boundary, and prove a uniqueness theorem. Finally, by using the method of sub- and super-solutions (order-preserving map iteration), we establish several existence results for a large class of problems covering a broad parameter regime, which includes most of the cases relevant in practice.
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