Semicontinuity and relaxation of L ∞-functionals

Prinari, Francesca

doi:10.1515/acv.2009.003

Cited by 13 publications

(15 citation statements)

References 6 publications

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“…Assuming that ⊂ R is an interval, they proved that F is sequentially L ∞ -weakly * lower semicontinuous if and only if the supremand f is level convex and lower semicontinuous. The same statement holds for general ⊂ R n ; see [1,Theorem 4.1], as well as [10,42].…”

Section: Supremal Functionals and Level Convexitymentioning

confidence: 63%

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Kreisbeck

Zappale

2020

Calc. Var.

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We study variational problems involving nonlocal supremal functionalswhere ⊂ R n is a bounded, open set and W : R m × R m → R is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for L ∞ -weak * lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak * closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

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Section: Supremal Functionals and Level Convexitymentioning

confidence: 63%

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Kreisbeck

Zappale

2020

Calc. Var.

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“…is a continuous function satisfying a linear growth condition, then f ∞ coincides with the greatest lower semicontinuous and level convex function which is less or equal to f (see [23,Corollary3.11]).…”

Section: Definition 12 a Functionmentioning

confidence: 99%

“…In order to apply the direct method of the calculus of variations the main issue is the lower semicontinuity of F . Semicontinuity properties for supremal functionals have recently been studied by many authors; we refer for instance to Barron-Jensen [6], Barron-Jensen-Wang [8], Prinari [22,23] and to the recent papers by Ansini-Prinari [2] and Ribeiro-Zappale [24]. In the context of supremal functionals, in [8] Barron, Jensen and Wang introduce the notion of the weak Morrey quasiconvexity as the natural extension of the notion of Morrey quasiconvexity (see [14] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On the lower semicontinuity and approximation of $${L^{\infty}}$$ L ∞ -functionals

Prinari

2015

Nonlinear Differ. Equ. Appl.

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In this paper we show that if the supremal functional F (V, B) = ess sup x∈B f (x, DV (x)) is sequentially weak* lower semicontinuous on W 1,∞ (B, R d) for every open set B ⊆ Ω (where Ω is a fixed open set of R N), then f (x, •) is rank-one level convex for a.e x ∈ Ω. Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the L p-approximation of a supremal functional F via Γ-convergence when f is a non-negative and coercive Carathéodory function.

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“…for every Σ 1 , Σ 2 ∈ M d×N and λ ∈ [0, 1]. In [24] Prinari removes the hypotheses (1.4) in the scalar case d = 1 and shows that (F p ) Γ-converges to the functionalF given bỹ In [9] Bocea and Nesi study the L p -approximation in the more general framework of A-quasiconvexity. More precisely, they consider the power-law functionals F p :…”

Section: +∞ Otherwisementioning

confidence: 99%

“…Moreover, since f is continuous, level convex and satisfies a linear growth condition, we have that lim p→∞ ((f p ) * * ) 1/p = f (see e.g. [24] Remark 3.12). Hence, passing to the limit as p → ∞ we get that in particular f = lim p→∞ f p ; i.e., f is A-∞ quasiconvex.…”

Section: Letmentioning

confidence: 99%

Power-Law Approximation under Differential Constraints

Ansini

Prinari

2014

SIAM J. Math. Anal.

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We study the Γ-convergence of the power-law functionals Fp(V) = Ω f p (x, V (x))dx 1/p , as p tends to +∞, in the setting of constant-rank operator A. We show that the Γ-limit is given by a supremal functional on L ∞ (Ω; M d×N) ∩ KerA where M d×N is the space of d × N real matrices. We give an explicit representation formula for the supremand function. We provide some examples and as application of the Γ-convergence results we characterize the strength set in the context of electrical resistivity.

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Semicontinuity and relaxation of L ∞-functionals

Abstract: Fixed a bounded open set of R N , we completely characterize the weak* lower semicontinuity of functionals of the form

Cited by 13 publications

References 6 publications

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

Lower semicontinuity and relaxation of nonlocal $$L^\infty $$-functionals

On the lower semicontinuity and approximation of $${L^{\infty}}$$ L ∞ -functionals

Power-Law Approximation under Differential Constraints

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