Electronic Journal of Differential Equations (Jan 2017)
A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity
Abstract
Let $u: \Omega \subseteq \mathbb{R}^n \to \mathbb{R}^N$ be a smooth map and $n,N \in \mathbb{N}$. The $\infty$-Laplacian is the PDE system $$ \Delta_\infty u :=\Big(Du \otimes Du + |Du|^2[Du]^\bot \otimes I\Big) :D^2u = 0, $$ where $[Du]^\bot := \hbox{Proj}_{R(Du)^\bot}$. This system constitutes the fundamental equation of vectorial calculus of variations in $L^\infty$, associated with the model functional $$ E_\infty (u,\Omega')= \big\| |Du|^2\big\|_{L^\infty(\Omega')} ,\quad \Omega' \Subset \Omega. $$ We show that generalised solutions to the system can be characterised in terms of the functional via a set of designated affine variations. For the scalar case N=1, we utilize the theory of viscosity solutions by Crandall-Ishii-Lions. For the vectorial case $N\geq 2$, we utilize the recently proposed by the author theory of $\mathcal{D}$-solutions. Moreover, we extend the result described above to the p-Laplacian, 1<p<infinity.
