On Minimisers of L^p-mean Distortion

We study the L^p-mean distortion functionals, (Formula presented.) for Sobolev self homeomorphisms of the unit disk (Formula presented.) with prescribed boundary values (Formula presented.) and pointwise distortion function (Formula presented.). Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of (Formula presented.) presenting results we know and making some conjectures. Remarkably, smooth minimisers of the (Formula presented.)-mean distortion functionals have inverses which are harmonic with respect to a metric induced by the distortion of the mapping. From this we are able to deduce that the complex conjugate Beltrami coefficient of a smooth minimiser is locally quasiregular and we identify the quasilinear equation it solves. This has other consequences such as a maximum principle for the distortion.