Journal of Differential Equations
We study C2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a ‘localized’ form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short-time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.
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Capogna, L., & Raich, A. (2012). An Aronsson type approach to extremal quasiconformal mappings. Journal of Differential Equations, 253(3), 851-877. http://dx.doi.org/10.1016/j.jde.2012.04.015