Tadeusz Iwaniec, Gaven Martin, Jani Onninen
Abstract: We study the $L^p$-mean distortion functionals, $E_p[f]=\iint K^p(z,f)dz$, $f_{|S}=f_0$ for Sobolev self homeomorphisms of the unit disk $D$ with prescribed boundary values $f_0:S\to S$ and pointwise distortion function $K=K(z,f)$. Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of Ep presenting results we know and making some conjectures. Remarkably, smooth minimisers of the $L^p$-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.
Journal: Computational Methods and Function Theory, April 2014
DOI: 10.1007/s40315-014-0063-1