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Stability of the $L^{p}$-Poincaré inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependency

Nurgissa Yessirkegenov, Amir Zhangirbayev

TL;DR

This paper proves stability for the $L^{p}$-Poincaré inequality under both Lebesgue and Gaussian measures by linking the deficit to the $L^{p}$-distance to the optimizer manifold $E_{Poin}$. The authors derive explicit, geometry-dependent lower bounds on convex domains of diameter $d$, showing a deficit bound of the form $\Delta(u) \ge c_{1}(p) \left(\frac{\pi_{p}}{d}\right)^{p} d(u,E_{Poin})^{p}$, with a Gaussian analogue for the corresponding optimizer set. The approach hinges on a $C_{p}$-identity for the deficit, the first eigenfunction of the $p$-Laplacian, and weighted log-concave Poincaré inequalities, leveraging Sakaguchi’s concavity result and Colesanti et al.'s geometric framework to obtain explicit geometric constants. Overall, the work provides concrete geometric constants and a rigorous pathway to stability in nonlinear Poincaré inequalities.

Abstract

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 3.6). 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.

Stability of the $L^{p}$-Poincaré inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependency

TL;DR

This paper proves stability for the -Poincaré inequality under both Lebesgue and Gaussian measures by linking the deficit to the -distance to the optimizer manifold . The authors derive explicit, geometry-dependent lower bounds on convex domains of diameter , showing a deficit bound of the form , with a Gaussian analogue for the corresponding optimizer set. The approach hinges on a -identity for the deficit, the first eigenfunction of the -Laplacian, and weighted log-concave Poincaré inequalities, leveraging Sakaguchi’s concavity result and Colesanti et al.'s geometric framework to obtain explicit geometric constants. Overall, the work provides concrete geometric constants and a rigorous pathway to stability in nonlinear Poincaré inequalities.

Abstract

In this paper, we obtain stability results for the -Poincaré inequality for both Lebesgue and Gaussian probability measures (Theorem 3.3 and Theorem 3.6). Our approach relies on properties of the first eigenfunction of the (Gaussian) -Laplacian operator and weighted Poincaré inequalities for log-concave measures on convex domains.
Paper Structure (3 sections, 12 theorems, 62 equations)

This paper contains 3 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Theorem 2.1

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n \ (n \ge 2)$ with smooth boundary $\partial\Omega$. Fix a number $p > 1$. Let $u \in W_0^{1,p}(\Omega)$ be a positive weak solution to the nonlinear eigenvalue problem Then, $v = \log u$ is a concave function.

Theorems & Definitions (18)

  • Theorem 2.1: sakaguchi1987concavity
  • Theorem 2.2: colesanti2025geometric
  • Proposition 2.3: colesanti2025geometric
  • Theorem 2.4: ferone2012remark
  • Definition 2.5
  • Lemma 2.6: cazacu2024hardy
  • Lemma 2.7: CT24
  • Lemma 2.8: CT24
  • Theorem 2.9: apseit2025sharp
  • Theorem 3.1: apseit2025sharp
  • ...and 8 more