A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909659790573568 |
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| author | Maso, Gianni Dal Donati, Davide |
| author_facet | Maso, Gianni Dal Donati, Davide |
| contents | We associate to every function $u\in GBD(Ω)$ a measure $μ_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $μ_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $μ^a_u$, $μ^c_u$, and $μ^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(Ω)$ functions. We then characterise the space $GSBD(Ω)$, originally defined only by slicing, as the space of functions $u\in GBD(Ω)$ such that $μ^c_u=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation Maso, Gianni Dal Donati, Davide Functional Analysis Analysis of PDEs 49Q20, 74A45 We associate to every function $u\in GBD(Ω)$ a measure $μ_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $μ_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $μ^a_u$, $μ^c_u$, and $μ^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(Ω)$ functions. We then characterise the space $GSBD(Ω)$, originally defined only by slicing, as the space of functions $u\in GBD(Ω)$ such that $μ^c_u=0$. |
| title | A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation |
| topic | Functional Analysis Analysis of PDEs 49Q20, 74A45 |
| url | https://arxiv.org/abs/2506.19978 |