The Lichnerowicz equation on compact manifolds with boundary
The Lichnerowicz equation on compact manifolds with boundary
复制标题
具有边界的紧流形上的 Lichnerowicz 方程
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
G. Tsogtgerel
中科院分区：
文献类型：
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作者：
M. Holst;G. Tsogtgerel
本文系统地研究了带边界紧致流形上爱因斯坦约束方程的适定性理论。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末端和/或陷阱表面的柯西曲面模型中。此外，在发展开的渐近欧氏流形的解理论时出现的一些技术障碍与带边界的紧致流形上的类似。作为第一步，我们在这里将自己限制在Lichnerowicz方程，也称为哈密顿约束方程，它是约束系统中非线性的主要来源。重点放在低规律性数据和不同类型的边界条件之间的相互作用上，这以前没有仔细分析过。为了建立一个反映已有闭流形理论的适定性理论，我们首先将Yamabe分类推广到具有边界的紧致流形上的非光滑度量。然后将共形不变性的一个结果推广到有边界的流形上，并证明了一个唯一性定理。最后，利用子解和上解的方法(保序映射迭代)，我们建立了一大类问题的几个存在性结果，这类问题覆盖了广泛的参数区间，包括了实际中的大多数情况。
In this paper 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 analysed 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.
影响因子：
3.5
作者：
S. Dain
通讯作者：
S. Dain