The objective of this article is to construct, from a bounded Lipschitz planar domain \(\Omega\) in \(\mathbb{R}^2\) into a Riemannian manifold \(N\), minimising \(2\)-harmonic maps in \(W^{1,2}(\Omega, N)\) with boundary condition \(g\in W^{1/2 , 2}(\partial \Omega, N)\), as the limit, when \(p\nearrow 2\), of minimising \(p\)-harmonic maps (\(p\in (1,2)\)) from \(\Omega\) to \(N\), with the same boundary values.
The direct method of Calculus of Variations will provide a minimiser of the Dirichlet energy unless the space of \(W^{1,2}\)-maps with boundary values \(g\) is empty, which is known to happen.
In particular, the authors prove that the \(p\)-energy of minimising \(p\)-harmonic maps with fixed boundary condition in \(W^{1/2 , 2}(\partial \Omega, N)\) converges, when \(p\nearrow 2\), to the singular energy \({\mathcal{E}}^{1,2}_{\mathrm{sg}}(g)\), which quantifies the free homotopy class of \(g\).
This leads to the convergence of a sub-sequence of minimising \(p_n\)-harmonic maps, still with boundary condition \(g\in W^{1/2 , 2}(\partial \Omega, N)\), to a renormalisable \(2\)-harmonic map, minimising the \(2\)-energy on \(\Omega\) minus a finite number of points and minimising, on the whole of \(\Omega\), the renormalised energy \[ {\mathcal{E}}^{1,2}_{\mathrm{ren}}(v) = \lim_{\rho \searrow 0} \int_{\Omega\setminus \cup_{i=1}^{k} {\mathbb B}^2(a_i,\rho)} \frac{|Dv|^2}{2} - {\mathcal{E}}^{1,2}_{\mathrm{sg}}(g)\log{\frac{1}{\rho}} \]
Instead of relying on the regularity of \(p\)-harmonic maps, which is always a tricky question, the method used is an adaptation of approaches employed for Ginzburg-Landau energies and \({\mathcal{E}}^{1,2}_{\mathrm{ren}}\) can be seen as a relaxation functional.