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 $u:\Omega \subseteq {\mathbb{R}}^n/!⟶{\mathbb{R}}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|H(Du)\big\|_{L^\infty(Ω)} . \] 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 $n=N\ge 2$ 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 $A(x):D^{2}u=0$ follows as a corollary. Hence, the celebrated $L^{\infty }$ scalar uniqueness theory of Jensen [J] has no counterpart when $N\ge 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x)\in \mathcal{K}\subseteq {\mathbb{R}}^{n\times n}$, $x\in \Omega$.

arXiv comment16 pages, 3 figures, to appear in Comm. on Pure Appl. Anal