Optimal ∞-Quasiconformal Immersions

Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, RG6 6AX, UK and BCAM, Alameda de Mazarredo 14, 48009 Bilbao, Spain.
n.katzourakis@reading.ac.uk

Received: 17 July 2013

Abstract

For a Hamiltonian K ∈ C²(R^{N × n}) and a map u:Ω ⊆ Rⁿ − → R^N, we consider the supremal functional (1)The “Euler−Lagrange” PDE associated to (1)is the quasilinear system (2)Here K_P is the derivative and [ K_P ] ^⊥ 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 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 [ K_P ] ^⊥ 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.