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

Abstract

Let $u: \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$ be a smooth map and $n,N \in \mathbb{N}$. The $\infty$-Laplacian is the PDE system \[ \tag{1} \label{1} \Delta_\infty u \, :=\, \Big(Du \otimes Du + |Du|^2[Du]^\bot\! \otimes I\Big) :D^2u\, =\, 0, \] where $[Du]^\bot := \text{Proj}_{R(Du)^\bot}$. \eqref{1} constitutes the fundamental equation of vectorial Calculus of Variations in $L^\infty$, associated to the model functional \[ \tag{2} \label{2} E_\infty (u,\Omega')\, =\, \big\| |Du|^2\big\|_{L^\infty(\Omega')} ,\ \ \ \Omega' \Subset \Omega. \] 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 $N\geq 2$, we utilise the recently proposed by the author theory of $\mathcal{D}$-solutions. Moreover, we extend the result described above to the $p$-Laplacian, $1<p<\infty$.