The paper proves log-concavity of the first p-eigenfunction with convex potentials, a sharp one-dimensional gap inequality for every p>1, and a higher-dimensional p=2 collapse dichotomy.
Stability of the $L^{p}$-Poincar\'e inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
In this paper, we obtain stability results for the $L^{p}$-Poincar\'e inequality for both Lebesgue measure and Gaussian probability measure (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 Poincar\'e inequalities for log-concave measures on convex domains.
fields
math.AP 1years
2026 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Fundamental Gaps for the Dirichlet \(p\)-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy
The paper proves log-concavity of the first p-eigenfunction with convex potentials, a sharp one-dimensional gap inequality for every p>1, and a higher-dimensional p=2 collapse dichotomy.