A unified approach to the weighted Grotzsch and Nitsche problems for mappings of finite distortion

This note deals with the existence and uniqueness of a minimiser of the following Grotzsch-type problem under some mild conditions, where F denotes the set of all homeomorphims f with finite linear distortion K(z, f) between two rectangles Q1 and Q2 taking vertices into vertices, I center dot is a positive, increasing and convex function, and lambda is a positive weight function. A similar problem of Nitsche-type, which concerns the minimiser of some weighted functional for mappings between two annuli, is also discussed. As by-products, our discussion gives a unified approach to some known results in the literature concerning the weighted Grotzsch and Nitsche problems.