Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in ${\mathrm{L}}^{\infty }$

Abstract

We study variational problems for second order supremal functionals ${\mathrm{F}}_{\infty }(u)=\Vert F(\cdot ,u,\mathrm{D}u,\mathrm{A}:{\mathrm{D}}^{2}u){\Vert }_{{\mathrm{L}}^{\infty }(\Omega )}$, where $F$ satisfies certain natural assumptions, $\mathrm{A}$ is a positive matrix, and $\Omega ⋐{\mathbb{R}}^n$. Higher order problems are very novel in the Calculus of Variations in ${\mathrm{L}}^{\infty }$, 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 ${\mathrm{F}}_{\infty }$. 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 ${\mathrm{F}}_{\infty }$; 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 $\partial \Omega$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$.

41 pages