| Abstract
| - For a Hamiltonian K ∈ C2(R N × n) and a map u:Ω ⊆ R n − → R N, we consider the supremal functional (1)\begin{equation} \label{1}% E_\infty (u,\Om) \ :=\ \big\|K(Du)\big\|_{L^\infty(\Om)}. end{equation}The “Euler −Lagrange” PDE associated to (1)is the quasilinear system (2)\begin{equation} \label{2} A_\infty u \, :=\, \left(K_P ot K_P + K[K_P]^\bot \! K_{PP}\right)(Du):D^2 u \, = \, 0. % end{equation}Here KP is the derivative and [ KP ] ⊥ is the projection on its nullspace. (1)and (2)are the fundamental objects of vector-valued Calculus of Variations in L∞ and first arose in recent work of the author [N. Katzourakis, J. Differ. Eqs. 253 (2012) 2123-2139; Commun. Partial Differ. Eqs. 39 (2014) 2091-2124]. Herein we apply our results to Geometric Analysis by choosing as K the dilation function \begin{eqnarray*} K(P)={|P|^2}{\det(P^\top\! P)^{-1/n}} end{eqnarray*}which measures the deviation of u from being conformal. Our main result is that appropriately defined minimisers of (1)solve (2). Hence, PDE methods can be used to study optimised quasiconformal maps. Nonconvexity of K and appearance of interfaces where [ KP ] ⊥ is discontinuous cause extra difficulties. When n = N, this approach has previously been followed by Capogna −Raich ? and relates to Teichmüller’s theory. In particular, we disprove a conjecture appearing therein.
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