Strictly Continuous Extension of Functionals With Linear Growth to the Space Bv

Rindler, Filip; Shaw, Giles

doi:10.1093/qmath/hav022

Cited by 15 publications

(25 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Otherwise, we give the following simple argument: assume that A is not FDN, so that the maps u j (x) = exp(jx · ξ)v lie in ker A for some non-zero complex ξ, v. We traced this example back to [44], but it was likely known before (cp. This in particular shows that BV A (B) ⊂ W k−1,n/(n−1) (B, V ), and it is likely that the inclusion is strictly continuous by an argument similar to [42,Prop. 3.7] in the BV-case.…”

Section: Comparison To the Bourgain-brezis Conditionmentioning

confidence: 92%

Embeddings for A-weakly differentiable functions on domains

Gmeineder

Raiță

2019

Journal of Functional Analysis

View full text Add to dashboard Cite

We prove that the inhomogeneous estimate of vector fields on balls in R n B |D k−1 u| n/(n−1) dxholds if and only if the linear, constant coefficient differential operator A of order k has finite dimensional null-space (FDN). This generalizes the Gagliardo-Nirenberg-Sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full-spaceproved by Van Schaftingen precisely for elliptic and cancelling (EC) operators, building on fundamental L 1 -estimates from the works of Bourgain and Brezis. We prove that FDN strictly implies EC and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level.2010 Mathematics Subject Classification. Primary: 46E35; Secondary: 26D10.provided that A is EC and u ∈ C ∞ (Ω, V ) satisfy B j u = 0 on ∂Ω, where B j is a (finite collection of) linear differential operator(s) defined on ∂Ω that satisfy the Lopatinskiȋ-Shapiro Complementing Conditions. Such a result would provide a reasonable analogue of the results in [33,1,2,27] to the case p = 1, in spite of Ornstein's Non-inequality. The aim of this paper is to confirm this expectation in the case when B j ≡ 0 ("no boundary condition") and Ω is a ball (whereas Van Schaftingen's result [56, Thm. 1.3] essentially deals with the antipodal case when B j = ∂ j ν , j = 0 . . . k − 1, i.e., "all boundary conditions"). We emphasize that in the present situation the geometry of ∂Ω is not the foremost problem, as is the extendibility of functions u : Ω → V to some v : R n → V while ensuring that Av ∈ L 1 (R n , V ) boundedly. In fact, as we shall see below, this property

show abstract

Section: Comparison To the Bourgain-brezis Conditionmentioning

confidence: 92%

Embeddings for A-weakly differentiable functions on domains

Gmeineder

Raiță

2019

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…This follows by an adaptation of [26,Prop. 3.7] originating in the concentration compactness principle from [21].…”

Section: Strict Density and Continuitymentioning

confidence: 99%

“…To complete the proof of Lemma 2.11, in contrast to [26,Prop. 3.7], we also need to deal with possible concentrations at infinity.…”

Section: Strict Density and Continuitymentioning

confidence: 99%

“…By our assumption, D k−j u m D k−j u in L n n−j , so we can extract a subsequence (not re-labelled) such that Strict convergence combined with Reshetnyak's Continuity Theorem [1,Thm 2.39] allow us to pass to the limit on the right-hand side. To deal with the left-hand side, we raise both sides to the power of n n−j and use the Brezis-Lieb Lemma [10] (up to subsequence), exactly as in [26,Prop. 3.7].…”

Section: Strict Density and Continuitymentioning

confidence: 99%

See 1 more Smart Citation

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

Raiță

Skorobogatova

2020

Calc. Var.

View full text Add to dashboard Cite

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 .

show abstract

“…To prove the upper bound, we will establish an area-strict density result in BH. The notion of area-strict convergence, as discussed in [40], is as follows. Definition 2.9.…”

Section: )mentioning

confidence: 99%

Relaxation of functionals in the space of vector-valued functions of bounded Hessian

Hagerty

2018

Calc. Var.

View full text Add to dashboard Cite

In this paper it is shown that if Ω ⊂ R N is an open, bounded Lipschitz set, and if f :is a continuous function with f (x, ·) of linear growth for all x ∈ Ω, then the relaxed functional in the space of functions of Bounded Hessian of the energyfor bounded sequences in W 2,1 is given byThis result is obtained using blow-up techniques and establishes a second order version of the BV relaxation theorems of Ambrosio and Dal Maso [2] and Fonseca and Müller [27]. The use of the blow-up method is intended to facilitate future study of integrands which include lower order terms and applications in the field of second order structured deformations.Mathematics Subject Classification (2010): 49J45, 49Q20

show abstract

Strictly Continuous Extension of Functionals With Linear Growth to the Space Bv

Cited by 15 publications

References 22 publications

Embeddings for A-weakly differentiable functions on domains

Embeddings for A-weakly differentiable functions on domains

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

Relaxation of functionals in the space of vector-valued functions of bounded Hessian

Contact Info

Product

Resources

About