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
For a Hamiltonian K ∈ C2(RN × n) and a map u:Ω ⊆ Rn − → RN, we consider the supremal functional (1)The “Euler−Lagrange” PDE associated to (1)is the quasilinear system (2)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 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.
Mathematics Subject Classification: 30C70 / 30C75 / 35J47
Key words: Quasiconformal maps / distortion / dilation / aronsson PDE / vector-valued calculus of variations inL∞ / ∞-Harmonic maps
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