The p-harmonic relaxation versus the Ginzburg-Landau functional

We will describe the manifold-valued harmonic extension 
problem of a boundary data defined on the boundary of a domain and 
taking values into the manifold. This extension has engineering 
applications, which we will present. Unfortunately, applying the direct 
method of calculus of variations to the Dirichlet energy can prove 
futile since the set of competitors can be empty for topological reasons 
that we will detail. We will then present the common points and 
differences between the two relaxations available on the market: the 
p-harmonic relaxation and the Ginzburg-Landau relaxation. This will be 
an opportunity to present and tease results obtained in collaboration 
with Bohdan Bulanyi and Jean Van Schaftingen.