English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free anddivergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \inW^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einsteinconstraint equations using the well known conformal method with data $\sigma$and $\tau$. We show that if no solution exists then there is a non-trivialsolution of another non-linear limit equation on $1$-forms. This last equationcan be shown to be without solutions no solution in many situations. As acorollary, we get existence of solutions of the vacuum Einstein constraintequation under explicit assumptions which in particular hold on a dense set ofmetrics $g$ for the $C^0$-topology.

A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method