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

In this paper, we prove the gradient stability of the $L^p$-Poincaré inequality with explicit constants on bounded convex domains for any $1<p<\mathrm{\infty }$. 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.