Energy maximum principle for vectorial higher order absolute minimisers in L^∞ and L^p

2026/06/30 by Simone Carano, Nikos Katzourakis, Roger Moser
Mathematics · #math.AP

paper · pdf

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We show that vectorial absolute minimisers of general k-th order supremal functionals in W^k,∞(Ω,\mathbb R^N) satisfy a maximum principle of the form max _U H (⋅, u, \mathrm D u, ..., \mathrm D^ku)=max_{∂ U}H (⋅, u, \mathrm D u, ..., \mathrm D^ku), ∀ U⊆Ω open, suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for p-harmonic maps for p<∞.

Citations

• An L^∞-variational problem involving the Fractional Laplacian
• Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order L^∞-variational problem
• On the minimisation of the Peak-to-average ratio
• Examples of p-harmonic maps
• Minimisers of supremal functionals and mass-minimising 1-currents
• Regularity from p-harmonic potentials to ∞-harmonic potentials in convex rings
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• Regularity of absolute minimizers for continuous convex Hamiltonians
• On the numerical approximation of p-Biharmonic and ∞-Biharmonic functions
• Minimal submanifolds

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• On the numerical approximation of vectorial absolute minimisers in L^∞
• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in L^∞