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

@misc{yessirkegenov2026explicit,
  title = {{Explicit sharp gradient stability of the $L^p$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator}},
  author = {Nurgissa Yessirkegenov and Amir Zhangirbayev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27281},
  eprint = {2610.27281},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27281},
}

TY  - UNPB
TI  - Explicit sharp gradient stability of the L^p-Poincaré inequality and fundamental gaps for the Dirichlet p-Laplacian operator
AU  - Nurgissa Yessirkegenov
AU  - Amir Zhangirbayev
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27281
UR  - https://arxiv.org/abs/2610.27281
AB  - In this paper, we establish sharp gradient stability of the L^p-Poincaré inequality with explicit constants on bounded convex domains for any 1<p<∞. 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. In the case 1<p<2, we also establish a sharp upper bound gradient stability estimate with an explicit constant and the optimal exponent p, matching the exponent obtained by Ignat and Zuazua [IZ26]. By extending the lower stability estimate to general bounded domains with C^2 boundary, we are able to confirm the conjecture posed by Fleckinger-Pellé and Takáč [FPT02, Section 8]. Finally, we apply the stability inequality to obtain positive fundamental gap estimates in terms of the inradius for the Dirichlet p-Laplacian.
ER  -