Optimal $\infty$-Quasiconformal Immersions

Katzourakis, Nikos

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

For a Hamiltonian $K\in C^2\left(R^{N\times n}\right)$ and a map $u:\Omega \subseteq R^n-\to R^N$, we consider the supremal functional

$$$$

(1)

The “Euler−Lagrange” PDE associated to (1)is the quasilinear system

$$A_{\infty }u:=\left(K_P{K}_P+K{\left[K_P\right]}^{\perp }{K}_{PP}\right)\left(Du\right):D^{2}u=0.$$

(2)

Here $K_P$ is the derivative and ${\left[K_P\right]}^{\perp }$ is the projection on its nullspace. (1)and (2)are the fundamental objects of vector-valued Calculus of Variations in $L^{\infty }$ 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

$$\begin{array}{c}\hfill K\left(P\right)={\left|P\right|}^{2}det{\left(P^{\top }P\right)}^{-1/n}\end{array}$$

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 ${\left[K_P\right]}^{\perp }$ 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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