A New Characterisation of $\mathrm{\infty }$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\mathrm{\infty }}$

Let $u:\mathrm{\Omega }\subseteq {\mathbb{R}}^n⟶{\mathbb{R}}^N$ be a smooth map and $n,N\in \mathbb{N}$. The $\mathrm{\infty }$-Laplacian is the PDE system $$\begin{array}{cccc}& \text{label}1{\mathrm{\Delta }}_{\mathrm{\infty }}u:=(Du\otimes Du+\mathrm{\mid }Du{\mathrm{\mid }}^2[Du{]}^{\mathrm{\perp }}\text{\hspace{0.17em}⁣}\otimes I):D^{2}u=0,& & \text{(1)}\end{array}$$ where $[Du]^\bot := $\text{Proj}$_{R(Du)^\bot}$. \eqref{1} constitutes the fundamental equation of vectorial Calculus of Variations in $L^{\mathrm{\infty }}$, associated to the model functional $$\begin{array}{cccc}& \text{label}2E_{\mathrm{\infty }}(u,{\mathrm{\Omega }}^{\mathrm{\prime }})=\parallel \mathrm{\mid }Du{\mathrm{\mid }}^2{\parallel }_{L^{\mathrm{\infty }}({\mathrm{\Omega }}^{\mathrm{\prime }})},{\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }\mathrm{.}& & \text{(2)}\end{array}$$ We show that generalised solutions to \eqref{1} can be characterised in terms of \eqref{2} via a set of designated affine variations. For the scalar case $N=1$, we utilise the theory of viscosity solutions of Crandall-Ishii-Lions. For the vectorial case $N\ge 2$, we utilise the recently proposed by the author theory of $\mathcal{D}$-solutions. Moreover, we extend the result described above to the $p$-Laplacian, $1<p<\mathrm{\infty }$.