This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj , Duj )j for (uj )j ⊂ BV(Ω; R m ) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalto the space BV(Ω; R m ). Lower semicontinuity results of this type were first obtained by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.
In this paper, we prove that the integral functional F[u] : BV(Ω; R m ) → R defined byis continuous over BV(Ω; R m ), with respect to the topology of area-strict convergence, a topol-dense. This provides conclusive justification for the treatment of F as the natural extension of the functional u →ˆΩ f (x, u(x), ∇u(x)) dx, defined for u ∈ W 1,1 (Ω; R m ). This result is valid for a large class of integrands satisfying |f (x, y, A)| ≤ C(1 + |y| d/(d−1) + |A|) and its proof makes use of Reshetnyak's Continuity Theorem combined with a lifting map µ[u] : BV(Ω; R m ) → M(Ω × R m ; R m×d ). To obtain the theorem in the case where f exhibits d/(d − 1) growth in the y variable, an embedding result from the theory of concentration-compactness is also employed.
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
We prove an integral representation theorem for the L 1 (Ω; R m )-relaxation of the functionalto the space BV(Ω; R m ) under very general assumptions, requiring principally that f be Carathéodory, partially coercive, and quasiconvex in the final variable. Our result is the first of its kind which applies to integrands which are unbounded in the u-variable and thus allows to treat many problems from applications. Such functionals are out of reach of the classical blow-up approach introduced by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49]. Our proof relies on an intricate truncation construction (in the x and u arguments simultaneously) made possible by the theory of liftings as introduced in the companion paper [23], and features techniques which could be of use for other problems featuring u-dependent integrands.
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