Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913928343191552 |
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| author | Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco |
| author_facet | Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco |
| contents | In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*}
\begin{cases}
\displaystyle - Δ_1 u= h(u)f & \text{in } Ω, \\ \newline
u=0 & \text{on } \partial Ω,
\end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(Ω)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity.
Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(Ω)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role.
The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco Analysis of PDEs In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - Δ_1 u= h(u)f & \text{in } Ω, \\ \newline u=0 & \text{on } \partial Ω, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(Ω)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(Ω)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator. |
| title | Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.13793 |