Counterexamples in calculus of variations in L∞ through the vectorial Eikonal equation

Abstract

We show that for any regular bounded domain Ω ⊆ R n , n = 2, 3, there exist infinitely many global diffeomorphisms equal to the identity on ∂Ω which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the ∞-Laplace system arising in vectorial Calculus of Variations in L ∞ does not suffice to characterise either limits of p-Harmonic maps as p → ∞, or absolute minimisers in the sense of Aronsson. RésuméNous montrons que pour tout domaine borné régulier Ω ⊆ R n , n = 2, 3, il existe une infinité de difféomorphismes globaux solutions de l'équation iconale,égauxà l'identité sur ∂Ω. Nous donnonségalement des exemples explicites de telles cartes dans des domaines annulaires. Ceci implique que le systéme du type ∞-Laplacien apparaissant dans le Calcul des Variations vectoriel dans L ∞ ne suffit pasà caractériser les limites pour p → ∞ des cartes p-harmoniques, ni les minimiseurs absolus au sens d'Aronsson. Contre-exemples dans le Calcul des Variations dans L ∞ par l'équation iconale vectorielle

On the numerical approximation of vectorial absolute minimisers in $$L^\infty $$

“…This is connected to that fact that, when N ≥ 2, the associated Euler-Lagrange ∞-Laplace system admits smooth non-minimising solutions which are characterised by two distinct variational concepts, see e.g. [3,30,36].…”

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]).…”

“…This owes to the fact that, unlike the scalar case, the Aronsson system admits arbitrarily smooth non-minimising solutions, even in the model case of the ∞-Laplacian. For details we refer to [37]. Since Aronsson's absolute minimisers do not characterise the Aronsson system, the natural question arises as to what is their PDE counterpart.…”

On the numerical approximation of vectorial absolute minimisers in $L^\infty$

“…Herein we present and perform numerical experiments based on a new method recently proposed by the first author in the joint papers [33,35], where various existence and uniqueness have been obtained. This approach is motivated by the paper [19] and relates to developments in mass transportation (see e.g.…”

“…However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when n, N ≥ 2. Herein we present numerical experiments based on a new method recently proposed by the first author in the papers [33,35].…”