Euler-Lagrange equations for variable-growth total variation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908325933744128 |
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| author | Górny, Wojciech Łasica, Michał Matsoukas, Alexandros |
| author_facet | Górny, Wojciech Łasica, Michał Matsoukas, Alexandros |
| contents | We consider a class of integral functionals with Musielak-Orlicz type variable growth, possibly linear in some regions of the domain. This includes $p(x)$ power-type integrands with $p(x)\ge 1$ as well as double-phase $p\!-\!q$ integrands with $p=1$. The main goal of this paper is to identify the $L^2$-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young's inequality. As an application, we obtain the Euler-Lagrange equation for the variant of the Rudin-Osher-Fatemi image denoising problem with variable growth regularising term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Euler-Lagrange equations for variable-growth total variation Górny, Wojciech Łasica, Michał Matsoukas, Alexandros Analysis of PDEs 35J60 (Primary) 35J92, 35J75, 35J20, 49J52 (Secondary) We consider a class of integral functionals with Musielak-Orlicz type variable growth, possibly linear in some regions of the domain. This includes $p(x)$ power-type integrands with $p(x)\ge 1$ as well as double-phase $p\!-\!q$ integrands with $p=1$. The main goal of this paper is to identify the $L^2$-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young's inequality. As an application, we obtain the Euler-Lagrange equation for the variant of the Rudin-Osher-Fatemi image denoising problem with variable growth regularising term. |
| title | Euler-Lagrange equations for variable-growth total variation |
| topic | Analysis of PDEs 35J60 (Primary) 35J92, 35J75, 35J20, 49J52 (Secondary) |
| url | https://arxiv.org/abs/2504.13559 |