Preprint

Inserted: 9 apr 2025
Last Updated: 9 apr 2025

Pages: 19
Year: 2025

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.

Keywords: Gamma convergence, Elliptic systems, Vectorial Calculus of Variations in $L^\infty$, higher order problems, Euler-Lagrange equations