Energy maximum principle for vectorial higher order absolute minimisers in $L^{\infty }$ and $L^p$

Abstract

We show that vectorial absolute minimisers of general $k$-th order supremal functionals in $W^{k,\infty }(\hat{I}\text{c◯},{\mathbb{R}}^N)$ satisfy a maximum principle of the form $$ \max_{\overline U} \mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big)=\max_{\partial U}\mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big), \qquad\forall\ U\subseteqΩ\mbox{ open}, $$ suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for $p$-harmonic maps for $p<\infty$.