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

The existence and uniqueness properties for extremal mappings with smallest weighted Lp distortion between annuli and the related Grötzsch‐type problems are discussed. An interesting critical phase‐type phenomenon is observed. When p1 minimizers always exist, regardless of the weight function. 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.