Abstract: The seminal work of Brezis-Coron (1983) for 2-dimensional harmonic maps introduces an estimate that leads to the existence of harmonic maps minimizing in different homotopy classes. This had important consequences dimension 3, leading to the celebrated proof by Tristan Rivière of the existence of everywhere discontinuous harmonic maps and the partial regularity result of Hardt-Lin-Poon for minimisers of the axially symmetric relaxed Dirichlet energy.
We will discuss what analogies and differences arise when following a similar path for the 1-dimensional half-harmonic map case and for the 4-dimensional Yang-Mills functional. The talk will be based on joint works with Ali Hyder (TIFR Bangalore) and Tristan Rivière (ETH Zurich), and is supported by Fondazione Cariplo and CDP.