Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order L ^∞ -variational problem

Abstract

We study a vectorial

-variational problem of second order, where the supremal functional depends on the vector function u through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser

u ∞ {u_{\infty}}

under prescribed Dirichlet boundary conditions, together with a characterisation of

u ∞ {u_{\infty}}

as solution of a specific system of PDEs. Our result can be seen as a twofold extension of the one in [N. Katzourakis and R. Moser, Existence, uniqueness and structure of second order absolute minimisers, Arch. Ration. Mech. Anal. 231 2019, 3, 1615–1634]: We generalise it to the vectorial setting and, at the same time, we consider more general elliptic operators in place of the Laplacian.