PreprintCC BY 4.0arXiv
From BV A to BV: An endpoint Korn estimate
Abstract
In this short note we prove that, given a ℂ-elliptic operator 𝒜 and a map u∈ BV𝒜(Ω;V) satisfying ∇apu∈ L1(Ω;V⊗ℝd), then u∈ BV(Ω;V). The result is quantitative and follows from the Korn-type estimate |Du|(Ω) ≤ Cd,𝒜( |∇apu|L1(Ω) + |𝒜u|(Ω) ), valid for every such u∈ BV𝒜(Ω;V). As a consequence, we obtain the characterization BV𝒜(Ω;V)∖ BV(Ω;V) = { u∈ BV𝒜(Ω;V): ∇apu∉ L1(Ω;V⊗ℝd) }. Generative AI has been exploited. The usage is detailed in a specific Section.
Related
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BibTeX
@misc{caroccia2026from,
title = {{From $\mathrmBV^\mathcal A$ to $\mathrmBV$: An endpoint Korn estimate}},
author = {Marco Caroccia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13103},
eprint = {2609.13103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13103},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From BV^ A to BV: An endpoint Korn estimate
AU - Marco Caroccia
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13103
UR - https://arxiv.org/abs/2609.13103
AB - In this short note we prove that, given a ℂ-elliptic operator 𝒜 and a map u∈ BV^𝒜(Ω;V) satisfying ∇_apu∈ L^1(Ω;V⊗ℝ^d), then u∈ BV(Ω;V). The result is quantitative and follows from the Korn-type estimate |Du|(Ω) ≤ C_d,𝒜( |∇_apu|_L^1(Ω) + |𝒜u|(Ω) ), valid for every such u∈ BV^𝒜(Ω;V). As a consequence, we obtain the characterization BV^𝒜(Ω;V)∖ BV(Ω;V) = { u∈ BV^𝒜(Ω;V): ∇_apu∉ L^1(Ω;V⊗ℝ^d) }. Generative AI has been exploited. The usage is detailed in a specific Section.
ER -