The Lichnerowicz equation on compact manifolds with boundary

TL;DR

This paper develops a well-posedness theory for the Lichnerowicz equation on compact manifolds with boundary, addressing boundary conditions and low regularity data relevant to general relativity and numerical simulations.

Contribution

It generalizes the Yamabe classification and conformal invariance to manifolds with boundary, establishing existence and uniqueness results for the Lichnerowicz equation.

Findings

Extended Yamabe classification to nonsmooth metrics with boundary

Proved conformal invariance for manifolds with boundary

Established existence and uniqueness results for broad boundary conditions

Abstract

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.