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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations -
N Katzourakis (2017)
A new characterisation of $$\infty $$ ∞ -harmonic and $$p$$ p -harmonic mappings via affine variations in $$L^\infty $$ L ∞Electron. J. Differ. Equ., 2017
- Publisher
- Springer Journals
- Copyright
- Copyright © 2019 by Springer Science+Business Media, LLC, part of Springer Nature
- Subject
- Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
- ISSN
- 0095-4616
- eISSN
- 1432-0606
- DOI
- 10.1007/s00245-019-09569-y
- Publisher site
- See Article on Publisher Site
Abstract
We discuss two distinct minimality principles for general supremal first order function- als for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson’s standard notion of absolute minimisers and the concept of ∞-minimal maps introduced more recently by the second author. We prove that C absolute minimisers characterise a divergence system with parame- ters probability measures and that C ∞-minimal maps characterise Aronsson’s PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson’s equation has an equivalent divergence counterpart. ∞ ∞ Keywords Calculus of variations in L · L variational principle · Aronsson system ·∞-Laplacian · Absolute minimisers ·∞-minimal maps Mathematics Subject Classification Primary 35J47 · 35J62 · 53C24; Secondary 49J99 1 Introduction 2 N N ×n n Let n, N ∈ N and H ∈ C × R × R with ⊆ R an open set. In this paper we consider the supremal functional 1,∞ E (u, O) := ess sup H(·, u, Du), u ∈ W (; R ), O , (1.1) loc Nikos Katzourakis has been partially financially supported by the EPSRC Grant No. EP/N017412/1. Nikos Katzourakis n.katzourakis@reading.ac.uk
Journal
Applied Mathematics and Optimization – Springer Journals
Published: Apr 24, 2019