Existence, uniqueness and characterisation of local minimisers in second order calculus of variations in L^∞

We study variational problems for second order supremal functionals F∞(u)=∥F(·,u,Du,A:D2u)∥L∞(Ω), where F satisfies certain natural assumptions, A is a positive symmetric 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.