This note deals with the existence and uniqueness of a minimiser of the following Grötzsch-type problem \documentclass[12pt]{minimal}
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\begin{document}$$\mathop {\inf }\limits_{f \in \mathcal{F}} \iint_{Q_1 } {\phi (K(z,f))\lambda (x)dxdy}$$\end{document} 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, ϕ is a positive, increasing and convex function, and λ 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 Grötzsch and Nitsche problems.
