Poincaré and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910597210177536 |
|---|---|
| author | Cruz-Uribe, David López-García, Fernando Ojea, Ignacio |
| author_facet | Cruz-Uribe, David López-García, Fernando Ojea, Ignacio |
| contents | We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinuous at every point of $Ω$. Our results follows from a local-to-global argument based on the continuity of certain Hardy type operators. We provide examples that show that our boundary $\log$-Hölder condition is essentially necessary for our main results. The same examples are adapted to show that this condition is not sufficient for other related inequalities. Finally, we give an application to a Neumann problem for a degenerate $p(\cdot)$-Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poincaré and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary Cruz-Uribe, David López-García, Fernando Ojea, Ignacio Analysis of PDEs Classical Analysis and ODEs Primary: 46E35, Secondary: 26D10, 42B35 We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinuous at every point of $Ω$. Our results follows from a local-to-global argument based on the continuity of certain Hardy type operators. We provide examples that show that our boundary $\log$-Hölder condition is essentially necessary for our main results. The same examples are adapted to show that this condition is not sufficient for other related inequalities. Finally, we give an application to a Neumann problem for a degenerate $p(\cdot)$-Laplacian. |
| title | Poincaré and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary: 46E35, Secondary: 26D10, 42B35 |
| url | https://arxiv.org/abs/2409.03660 |