Sharp remainder of the $L^{p}$-Poincaré inequality for Baouendi-Grushin vector fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913938782814208 |
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| author | Apseit, Kuralay Yessirkegenov, Nurgissa Zhangirbayev, Amir |
| author_facet | Apseit, Kuralay Yessirkegenov, Nurgissa Zhangirbayev, Amir |
| contents | In this paper, we establish a sharp remainder formula for the Poincaré inequality for Baouendi-Grushin vector fields in the setting of $L^{p}$ for complex-valued functions. In special cases, we recover previously known results. Consequently, we also derive the $L^{p}$-Poincaré inequality with an explicit optimal constant under a certain assumption. Additionally, we provide estimates of the remainder term for $p\geq2$ and $1<p<2\leq n<\infty$. As an application, we obtain a blow-up in finite time and global existence of the positive solutions to the initial-boundary value problem of the doubly nonlinear porous medium equation involving a degenerate nonlinear operator $Δ_{γ,p}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp remainder of the $L^{p}$-Poincaré inequality for Baouendi-Grushin vector fields Apseit, Kuralay Yessirkegenov, Nurgissa Zhangirbayev, Amir Analysis of PDEs 39B62, 35B44, 35A01 In this paper, we establish a sharp remainder formula for the Poincaré inequality for Baouendi-Grushin vector fields in the setting of $L^{p}$ for complex-valued functions. In special cases, we recover previously known results. Consequently, we also derive the $L^{p}$-Poincaré inequality with an explicit optimal constant under a certain assumption. Additionally, we provide estimates of the remainder term for $p\geq2$ and $1<p<2\leq n<\infty$. As an application, we obtain a blow-up in finite time and global existence of the positive solutions to the initial-boundary value problem of the doubly nonlinear porous medium equation involving a degenerate nonlinear operator $Δ_{γ,p}$. |
| title | Sharp remainder of the $L^{p}$-Poincaré inequality for Baouendi-Grushin vector fields |
| topic | Analysis of PDEs 39B62, 35B44, 35A01 |
| url | https://arxiv.org/abs/2507.01681 |