Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order $L^{\infty }$-variational problem

Abstract

We study a vectorial $L^{\infty }$-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 $u_{\infty }$ under prescribed Dirichlet boundary conditions, together with a characterisation of $u_{\infty }$ as solution of a specific system of PDEs. Our result can be seen as a twofold extension of the one in Katzourakis-Moser (ARMA 2019): we generalise it to the vectorial setting and, at the same time, we consider more general elliptic operators in place of the Laplacian.