Deformations with smallest weighted Lp average distortion and Nitsche-type phenomena

The existence and uniqueness properties for extremal mappings with smallest weighted L distortion between annuli and the related Grötzsch type problems are discussed. An interesting critical phase type phenomena is observed. When p < 1, apart from the identity map, minimizers never exist. When p = 1 we observe Nitsche type phenomena; minimisers exist within a range of conformal moduli determined by properties of the weight function and not otherwise. When p > 1 minimisers always exist. Interpreting the weight function as a density or “thickness profile” leads to interesting models for the deformation of highly elastic bodies and tearing type phenomena.