On the Structure of ∞-Harmonic Maps

Abstract

Let H ∈ C 2 (R N ×n ), H ≥ 0. The PDE system (1) A∞u := H P ⊗ H P + H[H P ] ⊥ H P P (Du) : D 2 u = 0 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 [K3]. The scalar case though has a long history initiated by Aronsson in [A1]. 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.

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.…”

Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

“…The Aronsson system was derived through the well-known method of L p -approximations and is being studied quite systematically since its discovery, see e.g. [28]- [31], [34,37]. The additional normal term which is not present in the scalar case imposes an extra layer of complexity, as it might be discontinuous even for smooth solutions (see [29,31]).…”

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

“…Up to the early 2010s, all considerations were restricted to the scalar case. The general vectorial case of (1.1) as well as the study of the associated PDE systems arising from more general first order functionals (1.5) has been initiated in a series of recent papers [20][21][22][23][24][25][26][27][28]. Besides the intrinsic mathematical interest of the field, L ∞ functionals can be applied in a variety of scenarios.…”

“…In the vectorial case, as in the scalar case, solutions of (1.1) may also be singular [20,22] but in addition to this further difficulties arise in the vectorial case that are not present in the scalar case. One such issue is that the projection Du ⊥ may be discontinuous even for C ∞ maps u, whence the nonlinear operator Δ ∞ of (1.2) may have discontinuous coefficients even when applied to C ∞ maps.…”