Stability of the $L^{p}$-Poincaré inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependence and applications to spectral gaps
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918362951450624 |
|---|---|
| author | Yessirkegenov, Nurgissa Zhangirbayev, Amir |
| author_facet | Yessirkegenov, Nurgissa Zhangirbayev, Amir |
| contents | In this paper, we obtain stability results for the $L^{p}$-Poincaré inequality for both Lebesgue and Gaussian probability measures (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover important results of Yu, Zhong [YZ86] and Smits [Smi96] (Corollary 3.9), related to the fundamental gap conjecture of the Laplacian (resolved by Andrews and Clutterbuck [AC11]), thereby providing an alternative proof. Moreover, we extend this spectral gap result to the $p$-Laplacian (Corollary 3.6). Such gap estimates for the Dirichlet $p$-Laplacian appear to be unavailable, as also observed in [DSW18]. Our approach relies on properties of the first eigenfunction of the (Gaussian) $p$-Laplacian operator and weighted Poincaré inequalities for log-concave measures on convex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of the $L^{p}$-Poincaré inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependence and applications to spectral gaps Yessirkegenov, Nurgissa Zhangirbayev, Amir Analysis of PDEs Probability 26D10, 35J60, 60E15 In this paper, we obtain stability results for the $L^{p}$-Poincaré inequality for both Lebesgue and Gaussian probability measures (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover important results of Yu, Zhong [YZ86] and Smits [Smi96] (Corollary 3.9), related to the fundamental gap conjecture of the Laplacian (resolved by Andrews and Clutterbuck [AC11]), thereby providing an alternative proof. Moreover, we extend this spectral gap result to the $p$-Laplacian (Corollary 3.6). Such gap estimates for the Dirichlet $p$-Laplacian appear to be unavailable, as also observed in [DSW18]. Our approach relies on properties of the first eigenfunction of the (Gaussian) $p$-Laplacian operator and weighted Poincaré inequalities for log-concave measures on convex domains. |
| title | Stability of the $L^{p}$-Poincaré inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependence and applications to spectral gaps |
| topic | Analysis of PDEs Probability 26D10, 35J60, 60E15 |
| url | https://arxiv.org/abs/2602.05968 |