Bounded weak solutions with Orlicz space data: an overview
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909359066316800 |
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| author | Cruz-Uribe, David |
| author_facet | Cruz-Uribe, David |
| contents | It is well known that non-negative solutions to the Dirichlet
problem $Δu =f$ in a bounded domain $Ω$, where $f\in L^q(Ω)$, $q>\frac{n}2$, satisfy
$\|u\|_{L^\infty(Ω)} \leq C\|f\|_{L^q(Ω)}$. We generalize
this result by replacing the Laplacian with a degenerate elliptic
operator, and we show that we can take the data $f$ in an Orlicz space
$L^A(Ω)$ that, in the classical case, lies strictly between $L^{\frac{n}{2}}(Ω)$ and
$L^q(Ω)$, $q>\frac{n}2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded weak solutions with Orlicz space data: an overview Cruz-Uribe, David Analysis of PDEs 35B45, 35D30, 35J25, 46E30 It is well known that non-negative solutions to the Dirichlet problem $Δu =f$ in a bounded domain $Ω$, where $f\in L^q(Ω)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(Ω)} \leq C\|f\|_{L^q(Ω)}$. We generalize this result by replacing the Laplacian with a degenerate elliptic operator, and we show that we can take the data $f$ in an Orlicz space $L^A(Ω)$ that, in the classical case, lies strictly between $L^{\frac{n}{2}}(Ω)$ and $L^q(Ω)$, $q>\frac{n}2$. |
| title | Bounded weak solutions with Orlicz space data: an overview |
| topic | Analysis of PDEs 35B45, 35D30, 35J25, 46E30 |
| url | https://arxiv.org/abs/2410.17054 |