2004
DOI: 10.1017/s0308210500003681
|View full text |Cite
|
Sign up to set email alerts
|

Strict convergence and minimal liftings in BV

Abstract: Given a function v ∈ BV (Ω; R m ), we introduce the notion of a minimal lifting of Dv. We prove that every v ∈ BV (Ω; R m ) has a unique minimal lifting, and we show that if v k → v strictly in BV , then the minimal liftings of v k converge weakly as measures to the minimal lifting of v. As an application, we deduce a result about weak continuity of the distributional determinant Det D 2 u with respect to strict convergence.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1
1

Citation Types

0
23
0

Year Published

2015
2015
2024
2024

Publication Types

Select...
7

Relationship

0
7

Authors

Journals

citations
Cited by 16 publications
(23 citation statements)
references
References 2 publications
0
23
0
Order By: Relevance
“…for every nonnegative function f ∈ C(Ω × R N × S N ×d−1 ). In [7] it was proven that if v n → v strictly in BV (Ω,…”
Section: Statement and Proof Of The Main Resultsmentioning
confidence: 99%
“…for every nonnegative function f ∈ C(Ω × R N × S N ×d−1 ). In [7] it was proven that if v n → v strictly in BV (Ω,…”
Section: Statement and Proof Of The Main Resultsmentioning
confidence: 99%
“…This paper also contains the first proofs of Lemma 3.2 and Proposition 3.15 below. Our proofs for these results are new and, in the case of Proposition 3.15, are obtained as a corollary of the Structure Theorem (Theorem 3.11), which does not feature in [28].…”
Section: Liftingsmentioning
confidence: 92%
“…Definition 3.1 was first given by Jung & Jerrard in [28] where the authors initiated the study of elementary liftings (which they refer to as minimal liftings), introduced below in Definition 3.5. This paper also contains the first proofs of Lemma 3.2 and Proposition 3.15 below.…”
Section: Liftingsmentioning
confidence: 99%
See 1 more Smart Citation
“…Our interest in liftings stems from the fact that, for positively 1-homogeneous integrands, they can be used to compute F and hence, after an application of Reshetnyak's Continuity Theorem, reduce the question of the strict continuity of F to that of the strict continuity of the map u → µ [u]. In this context, liftings were first defined and studied in [18], where the authors also note that strict continuity of the map u → µ[u] implies strict convergence of F for positively 1-homogeneous integrands. We will define liftings in a different (although equivalent) way and, as a consequence, provide a cleaner derivation of the properties of liftings that we require.…”
Section: Liftings and A Continuous Embeddingmentioning
confidence: 99%