Nonuniqueness in Vector-Valued Calculus of Variations in $L^\infty$ and some Linear Elliptic Systems

Abstract

For a Hamiltonian $H \in C^2(\mathbb{R}^{N \times n})$ and a map $u:\Omega \subseteq \mathbb{R}^n /!\longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,\Omega) \ :=\ \big\|H(Du)\big\|_{L^\infty(\Omega)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} \tag{2} A_\infty u := \Big(H_P \otimes H_P + H[H_P]^\bot /! H_{PP}\Big)(Du):D^2 u = 0. \] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [K1]. Herein we show that the Dirichlet problem for \eqref{2} admits for all $n=N\geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top /! P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [J] has no counterpart when $N\geq 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq \mathbb{R}^{n\times n}$, $x\in \Omega$.