On the numerical approximation of ∞ -harmonic mappings

Nikos Katzourakis; Tristan Pryer

doi:10.1007/s00030-016-0415-9

On the numerical approximation of ∞ -harmonic mappings

Nikos Katzourakis, Tristan Pryer

University of Reading

Research output: Contribution to journal › Article › peer-review

16 Citations (SciVal)

Abstract

A map u: Ω ⊆ Rⁿ⟶ ℝ^N, is said to be ∞-harmonic if it satisfies (Formula Presented.). 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 E_{∞(u, Ω) := ǁDuǁ}L^∞(Ω). 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.

Original language	English
Article number	61
Journal	Nonlinear Differential Equations and Applications
Volume	23
Issue number	6
DOIs	https://doi.org/10.1007/s00030-016-0415-9
Publication status	Published - 1 Dec 2016

Bibliographical note

Funding Information:
N.K. was partially supported through the EPSRC Grant EP/N017412/1. T.P. was partially supported through the EPSRC Grant EP/P000835/1.

Publisher Copyright:
© 2016, The Author(s).

Keywords

Interfaces
Phase separation
Vector-valued Calculus of Variations in L
∞-Laplacian

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1007/s00030-016-0415-9Licence: CC BY

Cite this

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title = "On the numerical approximation of ∞ -harmonic mappings",

abstract = "A map u: Ω ⊆ Rn⟶ ℝN, is said to be ∞-harmonic if it satisfies (Formula Presented.). 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 E∞(u, Ω) := ǁDuǁL∞(Ω). 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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AU - Pryer, Tristan

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N2 - A map u: Ω ⊆ Rn⟶ ℝN, is said to be ∞-harmonic if it satisfies (Formula Presented.). 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 E∞(u, Ω) := ǁDuǁL∞(Ω). 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.

AB - A map u: Ω ⊆ Rn⟶ ℝN, is said to be ∞-harmonic if it satisfies (Formula Presented.). 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 E∞(u, Ω) := ǁDuǁL∞(Ω). 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.

KW - Interfaces

KW - Phase separation

KW - Vector-valued Calculus of Variations in L

KW - ∞-Laplacian

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ER -

On the numerical approximation of ∞ -harmonic mappings

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this