Existence of Dim Solutions to the Equations of Vectorial Calculus of Variations in $L^\infty$

Nikos Katzourakis

Published 2015-02-04Version 1

In the very recent paper [K, ArXiv:1501.06164] we introduced a new duality-free theory of generalised solutions which applies to fully nonlinear PDE systems of any order. As one of our first applications, we proved existence of vectorial solutions to the Dirichlet problem for the $\infty$-Laplace PDE system which is the analogue of the Euler-Lagrange equation for the functional $E_\infty(u,\Omega)=\|Du\|_{L^\infty(\Omega)}$. Herein we prove existence of a solution $u:\Omega\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$ to the Dirichlet problem for the system arising from the functional $E_\infty(u,\Omega)=\|H(\cdot,u,u')\|_{L^\infty(\Omega)}$. This is nontrivial even in the present $1D$ case, since the equations are non-divergence, highly nonlinear, degenerate, do not have classical solutions and standard approaches do not work. We further give an explicit example arising in variational Data Assimilation to which our result apply.