Einstein-type elliptic systems

TL;DR

This paper studies semi-linear elliptic systems inspired by Einstein's constraint equations, proving existence results for solutions under various physical and geometric conditions on asymptotically Euclidean manifolds.

Contribution

It introduces a general existence theorem for Einstein-type elliptic systems and extends Helmholtz decomposition results on AE manifolds with boundary.

Findings

Existence of solutions for charged dust coupled to Einstein equations.

Solutions satisfying trapped surface conditions.

Extension of Helmholtz decomposition to AE manifolds with boundary.

Abstract

In this paper we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE) manifolds. In particular, electromagnetic fields give rise to this kind of system. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on AE manifolds with boundary, which extends and clarifies previously known results.