Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in $$L^\infty $$ L ∞
Nikos Katzourakis · Calculus of Variations and Partial Differential Equations · 2017
Consider the supremal functional 1 $$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$ applied to $$W^{1,\infty }$$ maps $$u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N$$ , $$N\ge 1$$ . Under certain assumptions on $$\mathscr {L}$$ , we prove for any given boundary data the existence of a map which is: 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.