Electron. J. Differential Equations, Vol. 2017 (2017), No. 29, pp. 1-19.

A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity

Nikos Katzourakis

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, $1<p<\infty$.

Submitted August 10, 2016. Published January 26, 2017.
Math Subject Classifications: 35D99, 35D40, 35J47, 35J47, 35J92, 35J70, 35J99.
Key Words: Infinity-Laplacian; p-Laplacian; generalised solutions; viscosity solutions; calculus of variations in L-infinity; Young measures; fully nonlinear systems.

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Nikos Katzourakis
Department of Mathematics and Statistics
University of Reading, Whiteknights, PO Box 220
Reading RG6 6AX, Berkshire, UK
email: n.katzourakis@reading.ac.uk

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