Characterization of generalized Young measures in the $mathcal{A}$-quasiconvexity context

Baía, Margarida; Matías, José; Santos, Pedro Miguel

doi:10.1512/iumj.2013.62.4928

Cited by 7 publications

(17 citation statements)

References 26 publications

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“…However, such properties are not known to hold for general higher-order operators. Up to now, the only Afree result in the generalized setting was a partial characterization due to Baía, Matias and Santos [11]. There, the authors characterize all generalized Young measures generated by A-free measures under the following somewhat restrictive assumptions: (a) The operator A is assumed to be of first-order.…”

Section: State Of the Artmentioning

confidence: 99%

“…which contains the set of B-gradients in Fourier space. The exactness property (11) has two direct consequences: First, it implies (see [54,Corollary 1]) the equivalence between A-quasiconvexity and B-gradient quasiconvexity:…”

Section: By(u ) ⊂ Y a (U )mentioning

confidence: 99%

“…The proof of the following proposition is contained in Lemma 5.3 of [11]. There the authors state their main results under additional assumptions.…”

Section: The Convexity Of Y A0 (μ )mentioning

confidence: 99%

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Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

Arroyo-Rabasa

2021

Arch Rational Mech Anal

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We give two characterizations, one for the class of generalized Young measures generated by $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -free measures and one for the class generated by $${\mathcal {B}}$$ B -gradient measures $${\mathcal {B}}u$$ B u . Here, $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A and $${\mathcal {B}}$$ B are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized $${\mathcal {A}}$$ A -free Young measures in duality with the class of $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized $${\mathcal {B}}$$ B -gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of $$\mathrm {L}^1$$ L 1 -compensated compactness when concentration of mass is allowed. These include the failure of $$\mathrm {L}^1$$ L 1 -estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $$\Omega $$ Ω , the inclusions $$\begin{aligned} \mathrm {L}^1(\Omega ) \cap \ker {\mathcal {A}}&\hookrightarrow {\mathcal {M}}(\Omega ) \cap \ker {{\,\mathrm{{\mathcal {A}}}\,}}\,,\\ \{{\mathcal {B}}u\in \mathrm {C}^\infty (\Omega )\}&\hookrightarrow \{{\mathcal {B}}u\in {\mathcal {M}}(\Omega )\} \end{aligned}$$ L 1 ( Ω ) ∩ ker A ↪ M ( Ω ) ∩ ker A , { B u ∈ C ∞ ( Ω ) } ↪ { B u ∈ M ( Ω ) } are dense with respect to the area-functional convergence of measures.

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Section: State Of the Artmentioning

confidence: 99%

Section: By(u ) ⊂ Y a (U )mentioning

confidence: 99%

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Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

Arroyo-Rabasa

2021

Arch Rational Mech Anal

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“…All of these arguments crucially use Alberti's rank-one theorem [1] (see [31] for a short and elegant new proof) and thus, since this theorem is specific to BV, extensions to further BV-like spaces have been prohibited so far. The only partial result for a characterization beyond BV seems to be in [7], but that result is limited to first-order operators (which does not cover BD) and also additional technical conditions have to be assumed.…”

Section: Introductionmentioning

confidence: 99%

Characterization of Generalized Young Measures Generated by Symmetric Gradients

Philippis

Rindler

2017

Arch Rational Mech Anal

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This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer-Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The "local" proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti's rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.

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“…Let us emphasize that variational problems with differential constraints naturally appear in hyperelasticity, electromagnetism, or in micromagnetics [7,25,26] and are closely related to the theory of compensated compactness [24,28,29]. The concept of A-quasiconvexity goes back to [5] and has been proved to be useful as a unified approach to variational problems with differential constraints, including results on homogenization [4,11], dimension reduction [19] and characterization of generalized Young measures [2] in the A-free setting. Moreover, first results on A-quasiaffine functions and weak continuity appeared in [16].…”

Section: Introductionmentioning

confidence: 99%

𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Kramer

Krömer

Kružík

et al. 2017

Advances in Calculus of Variations

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We state necessary and su cient conditions for weak lower semicontinuity of integral functionals of the form u → ∫ Ω h (x, u(x)) dx, where h is continuous and possesses a positively p-homogeneous recession function, p > , and u ∈ L p (Ω; ℝ m ) lives in the kernel of a constant-rank rst-order di erential operator A which admits an extension property. In the special case A = curl, apart from the quasiconvexity of the integrand, the recession function's quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly de ned notions of A-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in L p (Ω; ℝ m ) and approaching the kernel of A even if A does not have the extension property.

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Characterization of generalized Young measures in the $mathcal{A}$-quasiconvexity context

Cited by 7 publications

References 26 publications

Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

Characterization of Generalized Young Measures Generated by Symmetric Gradients

𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

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