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arXiv · 2507.01681

Sharp remainder of the $L^{p}$-Poincar\'e inequality for Baouendi-Grushin vector fields

Kuralay Apseit·Nurgissa Yessirkegenov·Amir Zhangirbayev
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Abstract

In this paper, we establish a sharp remainder formula for the Poincar\'e 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 $L^{p}$-Poincar\'e 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 $\Delta_{\gamma,p}$.

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Kuralay Apseit, Nurgissa Yessirkegenov, Amir Zhangirbayev. 2025-07-02. Sharp remainder of the $L^{p}$-Poincar\'e inequality for Baouendi-Grushin vector fields. https://arxiv.org/abs/2507.01681

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