分野別セミナー

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator of possibly non-integer order. They contain in particular harmonic and polyharmonic mappings. All these mappings are solutions to a quite general, critical PDE-system. I will present aspects of the regularity theory of this system which relies on the fine estimates from Harmonic Analysis and Potential Theory and the relation between antisymmetric potentials and integrability gains, a connection which was first observed in the 2-dimensional harmonic mapping case in T. Riviere's [Conservation laws for conformal invariant variational problems, Inventiones, 2007]. I will also mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying these analytic techniques.