On the divergence of the composition of irregular fields with BV functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910131960152064 |
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| author | Crasta, Graziano De Cicco, Virginia Malusa, Annalisa |
| author_facet | Crasta, Graziano De Cicco, Virginia Malusa, Annalisa |
| contents | We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14033 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the divergence of the composition of irregular fields with BV functions Crasta, Graziano De Cicco, Virginia Malusa, Annalisa Functional Analysis Analysis of PDEs 26B30, 49Q15, 49J45 We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces. |
| title | On the divergence of the composition of irregular fields with BV functions |
| topic | Functional Analysis Analysis of PDEs 26B30, 49Q15, 49J45 |
| url | https://arxiv.org/abs/2604.14033 |