2024/03/19 by Nikos Katzourakis, Katzourakis, Nikos, Roger Moser +1
Computer Science · Engineering · Mathematics · #35A15 #35B38 #35D99 #35J94 #49J27 #49K20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2403.12625
openalex publication_date 2024/03/19 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28
We study variational problems for second order supremal functionals \mathrm F_∞(u)= ‖F(⋅,u,\mathrm D u,A : \mathrm D2u)‖\mathrm L∞(Ω), where F satisfies certain natural assumptions, \mathrm A is a positive matrix, and Ω\Subset \mathbb Rn. Higher order problems are very novel in the Calculus of Variations in \mathrm L∞, 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 \mathrm F_∞. 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 \mathrm F_∞; 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 ∂ Ω, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if n=1.