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

TL;DR

This paper demonstrates the nonuniqueness of solutions in vector-valued calculus of variations in $L^ olinebreak \infty$ by constructing infinitely many solutions to associated PDEs, contrasting the scalar case where solutions are unique.

Contribution

It introduces a method to show nonuniqueness of solutions for vector-valued PDEs in $L^ olinebreak \\infty$, extending the understanding of calculus of variations in this setting.

Findings

Infinitely many smooth solutions exist for the Dirichlet problem in specific cases.

Nonuniqueness also applies to certain linear elliptic systems.

Scalar $L^ olinebreak \\infty$ theory of Jensen does not extend to vector-valued cases.

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 $Du(x) \in \mathcal{K} \subseteq \mathbb{R}^{n\times n}$, $x\in \Omega$.