Optimal regularity in a variational problem for current sheets in ideal magnetohydrodynamics

We characterize local behavior, and establish optimal local regularity, for minimizers of the functional $E(ψ)=∫Ω|∇ψ|2$ over collections 𝒞 that are weakly closed in $H1(Ω),$ closed under local smooth domain perturbations, and for which $E(ψ)$ controls $∫Ωψ2.$ Minimizers ψ satisfy a weak magnetohydrodynamic (MHD) equation and correspond to fields in low density ideal plasmas under cylindrical symmetry where the field component in the direction of the axis of symmetry is zero. We prove that $(∂ψ/∂x+i∂ψ/∂y)2$ is complex analytic, and locally $ψ=f(φ)$ for some φ, $f,$ with $Δφ=0$ and $f$ Lipschitz continuous with $|f′|=1$ almost everywhere, near points where $∇ψ≠0.$ An analogous but more elaborate characterization is established at points where $∇ψ=0.$ This characterization forms the basis for a general theory of the existence of current sheets due to imposed topological and boundary constraints. Results carry over to functions that are stationary points of $E(ψ)$ with respect to local smooth domain variations.