2016 # On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

**Abstract:** Abstract. A map u : Ω ⊆ R n −→ R N , is said to be ∞-harmonic if it satisfies(1)The system (1) is the model of vector-valued Calculus of Variations in L ∞ and arises as the "Euler-Lagrange" equation in relation to the supremal functional(2)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 occur…

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“…[9]). Since then, the field has been developed enormously by N. Katzourakis in the series of papers ( [11][12][13][14][15][16][17][18][19]) and also in collaboration with the author, Abugirda, Croce, Manfredi, Moser, Parini, Pisante and Pryer ( [1,2,6,[20][21][22][23][24][25]). A standard difficulty of (1.1) is that it is nondivergence form equation and since in general smooth solutions do not exist, the definition of generalised solutions is an issue.…”

confidence: 99%

“…[9]). Since then, the field has been developed enormously by N. Katzourakis in the series of papers ( [11][12][13][14][15][16][17][18][19]) and also in collaboration with the author, Abugirda, Croce, Manfredi, Moser, Parini, Pisante and Pryer ( [1,2,6,[20][21][22][23][24][25]). A standard difficulty of (1.1) is that it is nondivergence form equation and since in general smooth solutions do not exist, the definition of generalised solutions is an issue.…”

confidence: 99%

“…Since E p is a smooth functional, the minimizers can be computed using Newton's method. This concept was also pursued for the continuous setting in [25].…”

confidence: 99%

“…Diffuse derivatives can be seen as measure-theoretic disintegrations whose barycentres are the distributional derivatives (see [29]). For further results relevant to D-solutions and their applications, see [30]- [32], [34,10,35,13], [36]- [38].…”

confidence: 99%