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

We study a vectorial <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> {L∞} -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> </m:math> {u∞} under prescribed Dirichlet boundary conditions, together with a characterisation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> </m:math> {u∞} 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.