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, $1<p<\infty$.
Categories: math.AP