A Variational Shape Optimisation Approach to Multi-region Relaxed Magnetohydrodynamic Equilibria
Let $\Lambda \subset\mathbb{R}^3$ be a region admitting a partition into $n$ compact, connected subregions $\Lambda_1,\dots,\Lambda_n$, each with smooth boundary. Consider a vector field $B$ on $\Lambda$ where $B|_{\Lambda_i}$ is smooth, divergence free, and tangent to $\partial \Lambda_i$ for all $i$. We show that the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium equations are necessary and sufficient conditions for $ B $ and a metric to yield a stationary point of the magnetic energy under appropriate constraints. We constrain the pressure, relative helicity, and magnetic flux of $B$ through all smooth surfaces in $\Lambda_i$ whose boundary lies on $\partial \Lambda_i$. We identify a previously overlooked gauge condition. A definition for relative helicity is introduced, its gauge invariance is proved, and the existence of a gauge where relative helicity reduces to conventional helicity is demonstrated. In the case of a single region an additional condition is introduced that is sufficient to ensure a critical point of the magnetic energy is also a minimiser.