Universality of renormalisable mappings in two dimensions: the case of polar convex integrands

Abstract

We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form $\int_\Omega f(|\mathrm{D} u|)\,\mathrm{d} x$. The analysis relies on the condition that the vortex map ${x}/{\lvert x\rvert}$ has finite energy and that $t\mapsto f (\sqrt{t})$ is concave. We derive the leading order asymptotics and provide a detailed description of the convergence of $\mathrm{W}^{1,1}$-almost minimisers, leading to a characterization of second-order asymptotics. At the core of the method, we prove a ball merging construction (following Jerrard and Sandier's approach) for a general class of convex integrands. We therefore generalize the approximation by $p$-harmonic mappings when $p\nearrow 2$ and can also cover linearly growing functionals, including those of area-type.