Boundary Hardy inequality on functions of bounded variation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910019430121472 |
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| author | Adimurthi Roy, Prosenjit Sahu, Vivek |
| author_facet | Adimurthi Roy, Prosenjit Sahu, Vivek |
| contents | Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~Ω$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(Ω)$, $$\int_Ω \frac{|u(x)|^{p}}{δ^{p}_Ω(x)} dx \leq C\int_Ω |\nabla u(x) |^{p}dx,$$ where $δ_Ω(x)$ is the distance function from $Ω^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(Ω)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09823 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary Hardy inequality on functions of bounded variation Adimurthi Roy, Prosenjit Sahu, Vivek Analysis of PDEs 46E35, 26D15, 39B62 Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~Ω$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(Ω)$, $$\int_Ω \frac{|u(x)|^{p}}{δ^{p}_Ω(x)} dx \leq C\int_Ω |\nabla u(x) |^{p}dx,$$ where $δ_Ω(x)$ is the distance function from $Ω^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(Ω)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$. |
| title | Boundary Hardy inequality on functions of bounded variation |
| topic | Analysis of PDEs 46E35, 26D15, 39B62 |
| url | https://arxiv.org/abs/2405.09823 |