Minimizing N-harmonic maps with partially constrained boundary conditions of prescribed degree

In this talk we will focus, for N greater than or equal to 2, on fractional W^{1-1/N, N}-maps from an (N - 1)-dimensional sphere into itself as traces of maps with finite energies modelled on the N-Dirichlet integral. We consider energy minimization among maps with trace of prescribed topological degree. Our goal is to prove existence and quantitative stability for minimizers in homotopy classes. Due to the lack of compact Sobolev embeddings and conformal invariance, minimizing sequences may fail to converge strongly, leading to a concentration-compactness alternative and bubbling phenomena. When they occur, these concentration effects are investigated using PDE tools like blow-up analysis and measure-theoretic tools such as Cartesian currents. Finally, we discuss genericity results under perturbations of the energy functional and we present some open problems. This is a joint ongoing work with Xavier Lamy (Toulouse).