Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator

Nurgissa Yessirkegenov; Amir Zhangirbayev

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arXiv · 2610.27281

Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator

Nurgissa Yessirkegenov·Amir Zhangirbayev

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Abstract

In this paper, we prove the gradient stability of the $L^{p}$-Poincaré inequality with explicit constants on bounded convex domains for any $1<p<\infty$. These estimates control the distance to the optimizers in the strongest possible norm with the optimal stability exponent $\max\{2,p\}$, exhibiting the same phenomenon initially observed by Figalli and Zhang [FZ22a] for the $L^p$-Sobolev inequality. As a result, we also confirm the conjecture posed by Fleckinger-Pellé and Takáč [FPT02, Section 8]. We then apply the inequality to obtain positive fundamental gap estimates in terms of the inradius for the Dirichlet $p$-Laplacian.

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Nurgissa Yessirkegenov, Amir Zhangirbayev. 2026-10-19. Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator. https://arxiv.org/abs/2610.27281

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