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

Abstract

We prove an existence theorem for positive solutions to Lichnerowicztype equations on complete manifolds with boundary (M, ∂M, , ) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.

Einstein Type Systems on Complete Manifolds

“…In the above context, the main objective of this paper is to analyse existence results for the coupled Gauss-Codazzi system (4) on general complete manifolds, without a specific asymptotic structure. This follows the spirit of the work done by G. Albanese and M. Rigoli in [1,2], but now in the perspective of the developments commented above for the coupled system. As usual, this procedure will consist on two parts: first we prove a general existence criteria which relies on the existence of appropriate barrier functions (sub and supersolutions) and then we provide explicit constructions for these barriers.…”

“…Then, since φ ∈ W 2,p with p > n, we know from claim 1 that φ I ∈ H k . Then since q ∈ H s−2 , corollary 2.3 ensures that ∆ γ f = qφ 2) . Elliptic regularity then implies that f ∈ H min(k+1,s) .…”

“…Such a compact exhaustion follows from the completeness of the manifold and a classical regularization procedure to ensure the smoothness of the boundary (see remark 3.2. in [2]).…”

“…This last method is associated to the Einstein constraint equations (ECE) of general relativity (GR), which stand as necessary conditions for general relativistic initial data sets to admit a well-posed evolution problem [24,14]. 2 Let us briefly recall the setting of this last problem. To begin with, recall that within GR a space-time is defined as a Lorentzian manifolds (V n+1 , ḡ) satisfying the Einstein field equations 3…”

“…Since then several refinements have been obtained. In particular, the decoupled constraint system has been been analysed on non-compact manifolds, for instance on asymptotically Euclidean manifolds (AE) [17,48,54], asymptotically cylindrical (AC) manifolds [19,20], asymptotically hyperbolic (AH) manifolds [5,3] and, recently, the decoupled Lichnerowicz equation has been analysed on general complete manifolds in [1,2] where both existence and uniqueness results were obtained. Furthermore, low regularity results have been established for instance in [48,54,53,41,15,16] and through these papers some non-vacuum situations have been incorporated.…”