Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in L∞

We study variational problems for second order supremal functionals F_∞(u)= \|F(·,u, D u,A\!:\! D²u)\|_ L∞(Ω), where F satisfies certain natural assumptions, A is a positive matrix, and Ω Rⁿ. Higher order problems are very novel in the Calculus of Variations in L∞, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for F_∞. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for F_∞; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on ∂ Ω, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$.