The inhomogeneous Total Variation Flow with $L^1$-data
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915886943698944 |
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| author | Latorre, Marta de León, Sergio Segura |
| author_facet | Latorre, Marta de León, Sergio Segura |
| contents | This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the $1$--Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\timesΩ\,, \end{equation*} where $Ω\subset\R^N$ is a bounded open set with Lipschitz boundary, $u_0\in L^1(Ω)$ is the initial datum, and $f\in L_{loc}^1(0,+\infty; L^1(Ω))$ is the source term.
We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{$1$--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The inhomogeneous Total Variation Flow with $L^1$-data Latorre, Marta de León, Sergio Segura Analysis of PDEs 35K55, 35K20, 35K67, 35D30, 35A01, 35A02 This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the $1$--Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\timesΩ\,, \end{equation*} where $Ω\subset\R^N$ is a bounded open set with Lipschitz boundary, $u_0\in L^1(Ω)$ is the initial datum, and $f\in L_{loc}^1(0,+\infty; L^1(Ω))$ is the source term. We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{$1$--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case. |
| title | The inhomogeneous Total Variation Flow with $L^1$-data |
| topic | Analysis of PDEs 35K55, 35K20, 35K67, 35D30, 35A01, 35A02 |
| url | https://arxiv.org/abs/2603.23035 |