Optimal -Quasiconformal Immersions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 561-582.

For a Hamiltonian KC 2 (R N×n ) and a map u:ΩR n -R N , we consider the supremal functional

(1)
The “Euler−Lagrange” PDE associated to (1)is the quasilinear system
A u:=K P K P +K[K P ] K PP (Du):D 2 u=0.(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
K(P)=|P| 2 det(P P) -1/n
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.

DOI : 10.1051/cocv/2014038
Classification : 30C70, 30C75, 35J47
Mots clés : Quasiconformal maps, distortion, dilation, aronsson PDE, vector-valued calculus of variations inL, ∞-Harmonic maps
Katzourakis, Nikos 1

1 Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, RG6 6AX, UK and BCAM, Alameda de Mazarredo 14, 48009 Bilbao, Spain.
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Katzourakis, Nikos. Optimal $\infty{}$-Quasiconformal Immersions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 561-582. doi : 10.1051/cocv/2014038. http://archive.numdam.org/articles/10.1051/cocv/2014038/

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