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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.00545 |
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| _version_ | 1866917061843746816 |
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| author | Kałamajska, Agnieszka Peša, Dalimil Roskovec, Tomáš |
| author_facet | Kałamajska, Agnieszka Peša, Dalimil Roskovec, Tomáš |
| contents | We obtain the inequalities of the form $$\int_Ω|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_Ω \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +Θ,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(Ω)$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\cal T}_{H}(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot)$, which is the primitive of the weight $h(\cdot)$, and $Θ$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00545 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form Kałamajska, Agnieszka Peša, Dalimil Roskovec, Tomáš Analysis of PDEs 46E35, 46B70 We obtain the inequalities of the form $$\int_Ω|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_Ω \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +Θ,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(Ω)$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\cal T}_{H}(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot)$, which is the primitive of the weight $h(\cdot)$, and $Θ$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae. |
| title | Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form |
| topic | Analysis of PDEs 46E35, 46B70 |
| url | https://arxiv.org/abs/2308.00545 |