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In this paper we deal with the existence, regularity and Beltrami fieldproperty of magnetic energy minimisers under a helicity constraint. We inparticular tackle the problem of characterising local as well as globalminimisers of the given minimisation problem. Further we generalise Arnold'sresults concerning the problem of finding the minimum magnetic energy in anorbit of the group of volume-preserving diffeomorphisms to the setting ofabstract manifolds with boundary.

Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes