Relaxation for highly discontinuous, possibly unbounded, integral functionals
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917028054433792 |
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| author | Bertin, Tommaso Treu, Giulia |
| author_facet | Bertin, Tommaso Treu, Giulia |
| contents | We consider the functional \[ F(u)=\int_Ω f(\nabla u)\,dx\qquad u\inφ+W^{1,1}_0(Ω) \] where $Ω$ is a Lipschitz bounded open set of $\R^N$, $f:\R^N\to\R\cup \{+\infty\}$ is a superlinear Borel function, $φ\in W^{1,\infty}(Ω)$.
We prove that, if $f$ is superlinear and satisfies very weak assumptions, then the Lavrentiev phenomenon does not occur. We underline that our assumptions include the case of non continuous, non convex, and unbounded Lagrangians. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relaxation for highly discontinuous, possibly unbounded, integral functionals Bertin, Tommaso Treu, Giulia Analysis of PDEs 49-XX, 49jxx We consider the functional \[ F(u)=\int_Ω f(\nabla u)\,dx\qquad u\inφ+W^{1,1}_0(Ω) \] where $Ω$ is a Lipschitz bounded open set of $\R^N$, $f:\R^N\to\R\cup \{+\infty\}$ is a superlinear Borel function, $φ\in W^{1,\infty}(Ω)$. We prove that, if $f$ is superlinear and satisfies very weak assumptions, then the Lavrentiev phenomenon does not occur. We underline that our assumptions include the case of non continuous, non convex, and unbounded Lagrangians. |
| title | Relaxation for highly discontinuous, possibly unbounded, integral functionals |
| topic | Analysis of PDEs 49-XX, 49jxx |
| url | https://arxiv.org/abs/2510.17577 |