Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order $L^\infty$-variational problem
Simone Carano, Nikos Katzourakis, Roger Moser
TL;DR
The work addresses a vector-valued, second-order $L^\infty$ variational problem with a supremal functional $E_\infty(u)=\|F(\cdot, \mathrm{L}u)\|_{L^\infty}$, where $\mathrm{L}$ is a divergence-form elliptic operator. It develops an $L^p$-approximation and Gamma-convergence framework to prove existence and uniqueness of the global minimiser $u_\infty$ under Dirichlet data and derives a PDE system characterising $u_\infty$ via a limiting auxiliary field $f_\infty$, extending prior scalar results to the vectorial setting with general operators. The analysis combines sharp elliptic estimates, convex analysis of the integrand, and a strategic use of unique continuation to obtain nontrivial $f_\infty$ and a robust PDE characterization. The results yield that $u_\infty$ is the unique absolute minimiser for $E_\infty$ and satisfy a nonlocal-to-local PDE system that enforces $F(\cdot, \mathrm{L}u_\infty)=e_\infty$ a.e., with $\mathrm{L}f_\infty=0$, offering a rigorous framework for higher-order vector-valued $L^\infty$ problems and paving the way for further generalisations and applications in variational calculus under the $L^\infty$ norm.
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.