Some recent results about geometric variational problems

I will report on several results related to harmonic maps with free boundary/minimal surfaces with free boundary and their Ginzburg-Landau approximation. The common feature between these problems is their non locality. I will present the main features of the problems under consideration and develop a strategy on how to handle them. I will also present several open problems and draw several directions of research. If time permits, I will explain both elliptic and parabolic approaches.