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

Abstract

Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\, \|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq Ω, \] applied to $W^{1,\infty }$ maps $u:\Omega \subseteq \mathbb{R}⟶{\mathbb{R}}^N$, $N\ge 1$. 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 $p\to \infty$ for any $q\ge 1$ 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.}

Augmented version of a paper accepted for publication in Calculus of Variations and PDE, 28 pages

References in corpus (4)

• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$
• Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems
• $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^{\infty }$ via the singular value problem

Cited by in corpus (8)

• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$
• Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems
• On the Numerical Approximation of $\infty$-Harmonic Mappings
• Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^{\infty }$
• Vectorial variational principles in $L^{\infty }$ and their characterisation through PDE systems
• The Eigenvalue Problem for the $\infty$-Bilaplacian
• Phase separation of n dimensional infinity Harmonic mappings