Abstract

I plan to present a survey talk on regularity of harmonic and n-harmonic maps into compact Riemannian manifolds, putting the problem in a historical perspective and discussing known results, from the papers of R. Schoen and K. Uhlenbeck which appeared 40 years ago, through the work of F. Hélein on harmonic maps on planar domains, to the recent results of A. Schikorra and my Polish collaborators, M. Miśkiewicz and B. Petraszczuk.

I shall also present a new example by Petraszczuk who proved that a specific mildly nonlinear elliptic system in the plane (with a quadratic nonlinearity in the gradient) - considered already by J. Frehse in 1973 has the following property: given an arbitrary compact set K in the disc, there exists a solution which is discontinuous precisely on K, and smooth elsewhere. A few related open questions will be stated at the end.