Higher order generalizations of harmonic maps

Harmonic maps are one of the most famous geometric variational problems for maps between Riemannian manifolds. The harmonic map equation is a second order semilinear elliptic PDE and many results on questions of existence and qualitative behavior of solutions have been obtained in the literature.
Recently, many researchers got attracted in higher order variants of harmonic maps. In the first part of the talk we will give an overview and present some recent results on biharmonic maps which constitute a fourth order generalization of harmonic maps.
Finally, we will focus on a higher order version of harmonic maps which was initially proposed by Eells and Sampson in 1964 and present several recent results on the latter.This is joint work with Stefano Montaldo, Cezar Oniciuc and Andrea Ratto.