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

Simone Carano; Nikos Katzourakis; Roger Moser

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arXiv · 2504.04181

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

Simone Carano·Nikos Katzourakis·Roger Moser

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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.

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Simone Carano, Nikos Katzourakis, Roger Moser. 2025-04-05. Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order $L^\infty$-variational problem. https://arxiv.org/abs/2504.04181

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