Abstract
A map \(u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N\), is said to be \(\infty \)-harmonic if it satisfies
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The system (1) is the model of vector-valued Calculus of Variations in \(L^\infty \) and arises as the “Euler-Lagrange” equation in relation to the supremal functional
In this work we provide numerical approximations of solutions to the Dirichlet problem when \(n=2\) and in the vector valued case of \(N=2,3\) for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.
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N.K. was partially supported through the EPSRC Grant EP/N017412/1. T.P. was partially supported through the EPSRC Grant EP/P000835/1.
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Katzourakis, N., Pryer, T. On the numerical approximation of \(\infty \)-harmonic mappings. Nonlinear Differ. Equ. Appl. 23, 61 (2016). https://doi.org/10.1007/s00030-016-0415-9
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DOI: https://doi.org/10.1007/s00030-016-0415-9