Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^\infty$

TL;DR

This paper establishes the existence of absolutely minimizing solutions to vectorial calculus of variations in $L^ fty$, linking minimizers, generalized solutions, and $L^p$ limits, with partial regularity, motivated by meteorological data assimilation.

Contribution

It introduces a new approach to find vectorial absolute minimizers and generalized solutions for $L^ fty$ variational problems using $L^p$ approximation and $\mathcal{D}$-solutions.

Findings

Existence of vectorial absolute minimizers for the supremal functional.

Generalized solutions characterized via $\\mathcal{D}$-solutions.

Partial $C^2$ regularity off a nowhere dense set.

Abstract

Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\, \|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq \Omega, \] applied to $W^{1,\infty}$ maps $u:\Omega\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$, $N\geq 1$. Under certain assumptions on $L$, we prove for any given boundary data the existence of a map which is: i) a vectorial Absolute Minimiser of \eqref{1} in the sense of Aronsson, ii) a generalised solution to the ODE system associated to \eqref{1} as the analogue of the Euler-Lagrange equations, iii) a limit of minimisers of the respective $L^p$ functionals as $p\rightarrow\infty$ for any $q\geq 1$ in the strong $W^{1,q}$ topology \& iv) partially $C^2$ on $\Omega$ off an exceptional compact nowhere dense set. \noi {Our method is based on $L^p$ approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of $\mathcal{D}$-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.}