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 occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.

Explicit $$\infty$$ ∞ -harmonic functions in high dimensions

“…[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.…”

Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

“…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].…”

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

“…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].…”