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On the structure of infinity-harmonic maps

Katzourakis, N. (2014) On the structure of infinity-harmonic maps. Communications in Partial Differential Equations, 39 (11). pp. 2091-2124. ISSN 0360-5302

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To link to this item DOI: 10.1080/03605302.2014.920351

Abstract/Summary

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:47109
Publisher:Taylor & Francis

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