On the approximation of $SBD$ functions and some applications
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866916859359526912 |
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| author | Crismale, Vito |
| author_facet | Crismale, Vito |
| contents | Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$, and $SBD^p$, whose functions are in $SBD^p_\infty$ and the jump set has finite $\mathcal{H}^{n-1}$-measure. This generalises on the one hand the density result by [Chambolle, 2004-2005] and, on the other hand, extends in some sense the three approximation theorems in by [De Philippis, Fusco, Pratelli, 2017] for $SBV$, $SBV^p_\infty$, $SBV^p$ spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As application, the sharp version of two $Γ$-convergence results for energies defined on $SBD^2$ is derived. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_03076 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the approximation of $SBD$ functions and some applications Crismale, Vito Functional Analysis Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$, and $SBD^p$, whose functions are in $SBD^p_\infty$ and the jump set has finite $\mathcal{H}^{n-1}$-measure. This generalises on the one hand the density result by [Chambolle, 2004-2005] and, on the other hand, extends in some sense the three approximation theorems in by [De Philippis, Fusco, Pratelli, 2017] for $SBV$, $SBV^p_\infty$, $SBV^p$ spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As application, the sharp version of two $Γ$-convergence results for energies defined on $SBD^2$ is derived. |
| title | On the approximation of $SBD$ functions and some applications |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/1806.03076 |