A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^\infty$

Abstract

Let $n,N\in \mathbb{N}$ with $\Omega \subseteq \mathbb{R}^n$ open. Given $H \in C^2(\Omega \times \mathbb{R}^N\times \mathbb{R}^{Nn}),$ we consider the functional \[ \tag{1} \label{1} E_\infty (u,\mathcal{O})\, :=\, \underset{\mathcal{O}}{\mathrm{ess}\,\sup}\, H (\cdot,u,\mathrm{D} u) ,\ \ \ u\in W^{1,\infty}_\text{loc}(\Omega,\mathbb{R}^N),\ \ \ \mathcal{O} \Subset \Omega. \] The associated PDE system which plays the role of Euler-Lagrange equations in $L^\infty$ is \[ \label{2} \tag{2} \left\{ \begin{array}{r} H_{P}(\cdot, u, \mathrm{D}u)\, \mathrm{D} \big(H(\cdot, u, \mathrm{D} u)\big) \, = \, 0, \ \ \ H(\cdot, u, \mathrm{D} u) \, [\![H_{P}(\cdot, u, \mathrm{D} u)]\!]^\bot \Big(\mathrm{Div}\big(H_{P}(\cdot, u, \mathrm{D} u)\big)- H_{\eta}(\cdot, u, \mathrm{D} u)\Big)\, =\, 0, \end{array} \right. \] where $[\![A]\!]^\bot := \mathrm{Proj}_{R(A)^\bot}$. Herein we establish that generalised solutions to \eqref{2} can be characterised as local minimisers of \eqref{1} for appropriate classes of affine variations of the energy. Generalised solutions to \eqref{2} are understood as $\mathcal{D}$-solutions, a general framework recently introduced by one of the authors.