Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909456410869760 |
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| author | Bertin, Tommaso |
| author_facet | Bertin, Tommaso |
| contents | We consider the functional $$F_\infty(u)=\int_Ωf(x,u(x),\nabla u(x)) dx \quad\quad u\in φ+ W_0^{1,\infty}(Ω,\mathbb{R})$$ where $Ω$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $φ\in W^{1,\infty}(Ω)$. We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of $F_\infty$ both in $φ+W^{1,\infty}(Ω)$ and in the larger space $φ+W^{1,p}(Ω)$ can be represented by means of the bipolar $f^{**}$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty}(Ω)$ and $W^{1,1}(Ω)$ for autonomous Lagrangians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians Bertin, Tommaso Analysis of PDEs Functional Analysis 49J45 49J52 49K40 We consider the functional $$F_\infty(u)=\int_Ωf(x,u(x),\nabla u(x)) dx \quad\quad u\in φ+ W_0^{1,\infty}(Ω,\mathbb{R})$$ where $Ω$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $φ\in W^{1,\infty}(Ω)$. We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of $F_\infty$ both in $φ+W^{1,\infty}(Ω)$ and in the larger space $φ+W^{1,p}(Ω)$ can be represented by means of the bipolar $f^{**}$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty}(Ω)$ and $W^{1,1}(Ω)$ for autonomous Lagrangians. |
| title | Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians |
| topic | Analysis of PDEs Functional Analysis 49J45 49J52 49K40 |
| url | https://arxiv.org/abs/2501.08027 |