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Einstein-Type Elliptic Systems

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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.

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Notes

  1. See [16, Chapter 1] for a detailed derivation of these constraint equations.

  2. In order to achieve this result in the case of manifolds with boundary, one would need to complement the elliptic estimate of [40, Theorem 1.10] with the corresponding estimates on the compact core, which follows along the same lines to that of [10, Proposition 4].

  3. Following [40], the exceptional integers are defined as \(\{z\in {\mathbb {Z}}, z\ne -1,\cdots ,3-n \}\).

  4. Here, we are using [44, Theorem 3.1] and the same observation applied in [38, Theorem 4.1]. That is, the hypotheses of [44, Theorem 3.1] can be weakened so as to suppose \(u\in W^p_{k,loc}\) instead of \(W^p_{k}\).

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Acknowledgements

Rodrigo Avalos would like to thank CAPES/PNPD and FUNCAP for financial support, and Jorge H. Lira would like to thank CNPq and FUNCAP for financial support. Also, we would like to thank the comments, suggestions and critics made be two anonymous referees which have helped us improve the content and presentation of the paper.

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  1. Department of Mathematics, Federal University of Ceará, Campus do Pici – Bloco 914, Fortaleza, Ceará, 60440-900, Brazil

    Rodrigo Avalos & Jorge H. Lira

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  1. Rodrigo Avalos
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  2. Jorge H. Lira
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Correspondence to Rodrigo Avalos.

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Communicated by Mihalis Dafermos.

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Avalos, R., Lira, J.H. Einstein-Type Elliptic Systems. Ann. Henri Poincaré 23, 3221–3264 (2022). https://doi.org/10.1007/s00023-022-01180-2

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