On the numerical approximation of vectorial absolute minimisers in L^infty

Abstract

Let Omega be an open set. We consider the supremal functional 1E∞(u,O):=‖Du‖L∞(O),O⊆Ωopen,\\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}$$\\begin{aligned} \\text {E}_\\infty (u,{\\mathcal {O}})\\, {:}{=}\\, \\Vert \\text {D}u \\Vert _{L^\\infty ( {\\mathcal {O}} )}, \\ \\ \\ {\\mathcal {O}} \\subseteq \\Omega \\text { open}, \\end{aligned}$$\\end{document}applied to locally Lipschitz mappings u : mathbb {R}^n supseteq Omega longrightarrow mathbb {R}^N, where n,Nin mathbb {N}. This is the model functional of Calculus of Variations in L^infty . The area is developing rapidly, but the vectorial case of Nge 2 is still poorly understood. Due to the non-local nature of (1), usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when n,Nge 2. We present numerical experiments aimed at understanding the behaviour of minimisers through a new technique involving p-concentration measures.