Some results on the relaxation of the area functional in dimension two and codimension two

We discuss some results on the value of the lower semicontinuous envelope of the area functional for graphs, in dimension two and codimension two. Namely, given a nonsmooth map $u$ from the plane to the plane, for instance a piecewise constant map – or more generally a map with some kind of singularities – we show some estimates for the value of the functional measuring the relaxed area of the graph of $u$, a nonsmooth two-dimensional manifold in four-dimensional space. Geometrically, the problem is to understand how to fill the "holes" in the graph of $u$, in the most economic way in terms of two-dimensional area, using sequences of approximating smooth two-dimensional surfaces of graph type in $\mathbb{R}^4$. Difficulties are due to the codimension two; the choice of the convergence with respect to which one decides to relax the area functional is also of importance.