A new space of generalised vector-valued functions of bounded variation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910000195043328 |
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| author | Donati, Davide |
| author_facet | Donati, Davide |
| contents | In [Dal Maso and Toader, NoDEA 2022], the authors introduced the space $GBV_\star(A)$ to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of $GBV_\star(A)$ to the vectorial case, introducing the space $GBV_\star(A;\mathbb{R}^k)$. We study the main properties of $GBV_\star(A;\mathbb{R}^k)$ and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [Dal Maso and Toader, NoDEA 2022] to show that minimising sequences in $GBV_\star(A;\mathbb{R}^k)$ can be modified to obtain a minimising sequence converging $\mathcal{L}^d$-a.e in $A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_11538 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new space of generalised vector-valued functions of bounded variation Donati, Davide Analysis of PDEs Classical Analysis and ODEs 26A45, 49J45 In [Dal Maso and Toader, NoDEA 2022], the authors introduced the space $GBV_\star(A)$ to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of $GBV_\star(A)$ to the vectorial case, introducing the space $GBV_\star(A;\mathbb{R}^k)$. We study the main properties of $GBV_\star(A;\mathbb{R}^k)$ and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [Dal Maso and Toader, NoDEA 2022] to show that minimising sequences in $GBV_\star(A;\mathbb{R}^k)$ can be modified to obtain a minimising sequence converging $\mathcal{L}^d$-a.e in $A$. |
| title | A new space of generalised vector-valued functions of bounded variation |
| topic | Analysis of PDEs Classical Analysis and ODEs 26A45, 49J45 |
| url | https://arxiv.org/abs/2310.11538 |