The Lichnerowicz equation on compact manifolds with boundary

Abstract

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.

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

“…For further informations on the initial value problem for the Einstein equations we refer to the recent surveys [5,7,11], and the references therein. The scalar equation in (1.2) is called the Lichnerowicz equation; since it is the main source of nonlinearity in the system (1.2), a good understanding of its solutions is a crucial step toward the resolution of the Einstein equations by the conformal method, see for instance [7,8,10,18,25,21,22,26,27,29,30,33,35,35]. Here we generalize many of the results obtained in the above papers, especially relaxing the assumptions on M .…”

“…Another natural issue in this framework, inspired by the classical singularity theorems of Hawking and Penrose, is to understand the behaviour of an initial data set containing event horizons. The approach introduced in [40] and which has been highly developed in recent years, see for instance [24,26,32], consists in excising the regions containing black holes and coherently impose some boundary conditions on the conformal factor. The boundary conditions introduced in the aforementioned references can be gathered into the following type of conditions (1.3) ∂νu + gH,ju + g θ,j u e j + gτ,ju N/2 + gw,j u −N/2 = 0 on ∂jM , for each boundary component ∂jM ; where the coefficients g·,j and ej, are related to the physical meaning of that boundary component, see [26] for a comprehensive exposition of the problem.…”

“…so that the modulation b θ (x) can be interpreted phisically as a smallness requirement for the positive part U+(ψ) of the potential. Concerning the boundary conditions, the restrictions on gi(x) and qi that we have in mind are this last condition in the literature of mathematical General Relativity is known as the defocusing case and it is meaningful for the applications, see for instance the aforementioned [26,24] and the references therein.…”

Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries

“…In order to obtain solutions to (1.1)-(1.2) satisfying the marginally trapped surface conditions, we impose boundary conditions on (ĝ ab ,K ab ) over Σ. Following the discussion in [9] and [8], a marginally trapped surface is one whose expansion along the incoming and outgoing orthogonal, null geodesics is non-positive. On the boundary Σ, the expansion scalars are given bŷ θ ± = ∓(n − 1)Ĥ + trĝK −K(ν,ν), (1.3) where (n − 1)Ĥ = divĝν is the mean extrinsic curvature of Σ andν is the outward pointing, unit normal vector field to M. Therefore, the surface Σ is called a marginally trapped surface ifθ ± 0.…”

“…(1. 8) In (1.6), H is the rescaled extrinsic curvature for the boundary, ν = φ N 2 −1ν is the rescaled normal vector field, and d n = n−2 2 is a dimension dependent constant. The operators ∂ ν and L are defined with respect to the specified metric g. In order to guarantee that θ + 0, the scalar function ψ is chosen so that φ ψ.…”