Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

Arroyo-Rabasa, Adolfo; Philippis, Guido De; Hirsch, Jonas; Rindler, Filip

doi:10.1007/s00039-019-00497-1

Cited by 39 publications

(59 citation statements)

References 45 publications

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“…4.1.3]), and the last estimate follows from Hölder's inequality. We next note that by the dominated convergence theorem, we have that |Bu|(B 2r (x)) → |Bu|({x 0 }) as r ↓ 0, which is null, e.g., by [3]. By (1.1) and the density lemma, we have that D n−j u ∈ L n/(n−j) loc (Ω), so that the summands also tend to zero by dominated convergence theorem.…”

Section: Canceling Operatorsmentioning

confidence: 99%

Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

Raiță

Skorobogatova

2020

Calc. Var.

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We prove that for elliptic and canceling linear differential operators B of order n on R n , continuity of a map u can be inferred from the fact that Bu is a measure. We also prove strict continuity of the embedding of the space BV B (R n ) of functions of bounded B-variation into the space of continuous functions vanishing at infinity.2010 Mathematics Subject Classification. Primary: 26D10; Secondary: 46E35 .

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Section: Canceling Operatorsmentioning

confidence: 99%

Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

Raiță

Skorobogatova

2020

Calc. Var.

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“…In two dimensions, this setting is associated to the operator A(w 1 , w 2 ) = (∂ 2 w 1 , ∂ 1 w 2 ), which is one of the simplest examples of an operator that does not satisfy the constant rank property (see [63]). Other related work about the understanding of PDE-constraints where the constant rank property is not a main assumption include [7,8,22,65], and more recently [9].…”

Section: Comments On the Constant Rank Assumptionmentioning

confidence: 99%

“…A direct consequence of this lemma is the following failure of the L 1 -rigidity for elliptic systems (cf. [9] and [48, Sect. 2.6]) for A-free measures.…”

Section: Failure Of L 1 -Compactness For Elliptic Systemsmentioning

confidence: 99%

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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“…These operators have been recently considered in e.g. [17,46,47,37,10,57]. The deviator operator E D u := Eu − 1 n (div u) Id n is C-elliptic for n ≥ 3, but not for n = 2 (see Remark 2.5).…”

Section: Introductionmentioning

confidence: 99%

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

Chambolle

Crismale

2020

Advances in Calculus of Variations

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We study the Γ-limit of Ambrosio-Tortorelli-type functionals Dε(u, v), whose dependence on the symmetrised gradient e(u) is different in Au and in e(u) − Au, for a C-elliptic symmetric operator A, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano in [39]. In particular we prove that G(S)BD functions with bounded A-variation are (S)BD.

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Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

Cited by 39 publications

References 45 publications

Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

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

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

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