Existence, uniqueness and characterisation of local minimisers in second-order calculus of variations in L∞

Katzourakis, Nikos; Moser, Roger

doi:10.2140/apde.2026.19.1251

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Existence, uniqueness and characterisation of local minimisers in second-order calculus of variations in $\mathrm{L}^{\infty}$

Nikos Katzourakis and Roger Moser

Vol. 19 (2026), No. 7, 1251–1294

DOI: 10.2140/apde.2026.19.1251

Abstract

We study variational problems for second-order supremal functionals $F_{\infty} (u) = ∥ F (\cdot, u, D u, A : D^{2} u) ∥_{L^{\infty} (Ω)}$ , where $F$ satisfies certain natural assumptions, $A$ is a positive symmetric matrix, and $Ω ⋐ ℝ^{n}$ . Higher-order problems are very novel in the calculus of variations in $L^{\infty}$ and exhibit a strikingly different behaviour compared to first-order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $F_{\infty}$ . We prove that, under appropriate conditions, “localised” minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $F_{\infty}$ ; the latter is only a necessary but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial Ω$ , as well as their partial regularity outside a singular set of codimension $1$ , which may be nonempty even if $n = 1$ .

Keywords

calculus of variations in $\mathrm{L}^{\infty}$, higher-order problems, local minimisers, absolute minimisers, Euler–Lagrange equations, Aronsson equations

Mathematical Subject Classification

Primary: 35A15, 35B38, 35D99, 35J94, 49K20

Secondary: 49J27

Milestones

Received: 17 June 2024

Revised: 16 July 2025

Accepted: 19 September 2025

Published: 18 September 2026

Authors

Nikos Katzourakis
Department of Mathematics and Statistics University of Reading Reading United Kingdom

Roger Moser
Department of Mathematical Sciences University of Bath Bath United Kingdom

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Vol. 19, No. 7, 2026