Abstract
Consider the supremal functional 1E∞(u,A):=‖L(·,u,Du)‖L∞(A),A⊆Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\infty (u,A) := \\Vert \\mathscr {L}(\\cdot ,u,\\mathrm {D}u)\\Vert _{L^\\infty (A)},\\quad A\\subseteq \\Omega , \\end{aligned}$$\\end{document}applied to W^{1,infty } maps u:Omega subseteq mathbb {R}longrightarrow mathbb {R}^N, Nge 1. Under certain assumptions on mathscr {L}, we prove for any given boundary data the existence of a map which is:a vectorial Absolute Minimiser of (1) in the sense of Aronsson,a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations,a limit of minimisers of the respective L^p functionals as prightarrow infty for any qge 1 in the strong W^{1,q} topology andpartially C^2 on Omega off an exceptional compact nowhere dense set. Our method is based on L^p approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of mathcal {D}-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.
Highlights
Calculus of Variations in L∞ has a long history and was pioneered by Aronsson in the 1960s [4,5,6,7,8]
An extra difficulty of (1.5) which is not present in the scalar case of (1.3) is that the coefficients may be discontinuous along interfaces even for C∞ solutions because the term involving [Du]⊥ measures the dimension of the tangent space of u( ) ⊆ RN
Motivated in part by the nonlinear systems arising in L∞, in the very recent paper [39] the author proposed a new theory of generalised solutions which applies to fully nonlinear PDE systems
Summary
Calculus of Variations in L∞ has a long history and was pioneered by Aronsson in the 1960s [4,5,6,7,8]. An extra difficulty of (1.5) which is not present in the scalar case of (1.3) is that the coefficients may be discontinuous along interfaces even for C∞ solutions because the term involving [Du]⊥ measures the dimension of the tangent space of u( ) ⊆ RN (see [32,33] and the numerical experiments in [43]) This is a general vectorial phenomenon studied in some detail in [34]. Let us note that, inspired by Aronsson’s work, Sheffield-Smart [52] made a vectorial breakthrough relevant to (1.5) and (1.4) which was simultaneous to [32] They studied smooth vectorvalued optimal Lipschitz extensions of functions, deriving a different more singular version of ∞-Laplacian than (1.5), corresponding to (1.4) but when the matrix norm of Du is the nonsmooth operator norm on RN×n. Our main result establishes for any given endpoint data on the existence
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