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

TL;DR

This paper proves the universality of renormalised energy for planar mappings with convex integrands, establishing asymptotic behavior and convergence of almost minimisers, extending previous approximation methods.

Contribution

It introduces a ball merging construction for convex integrands, generalizing approximation techniques for mappings with various growth conditions.

Findings

Established universality of renormalised energy in 2D mappings.

Derived leading order asymptotics and convergence of minimisers.

Extended approximation methods to linearly growing functionals.

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.