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Maz'ya-type bounds for sharp constants in fractional Poincaré-Sobolev inequalities

Francesco Bozzola, Matteo Talluri

Abstract

We prove estimates for the sharp constants in fractional Poincaré-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local case by Maz'ya and Shubin and by the first author and Brasco. We rely on a new Maz'ya-Poincaré inequality and, incidentally, we also prove new fractional Poincaré-Wirtinger-type estimates. These inequalities display sharp limiting behaviours with respect to the fractional order of differentiability. As a byproduct, we obtain a new criterion for the embedding of the homogeneous Sobolev space $\mathcal{D}^{s,p}_0(Ω)$ in $L^q(Ω)$, valid in the subcritical regime and for $p \le q < p^*_s$. Our results are new even for the first eigenvalue of the fractional Laplacian and contain an optimal characterization for the positivity of the fractional Cheeger's constant.

Maz'ya-type bounds for sharp constants in fractional Poincaré-Sobolev inequalities

Abstract

We prove estimates for the sharp constants in fractional Poincaré-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local case by Maz'ya and Shubin and by the first author and Brasco. We rely on a new Maz'ya-Poincaré inequality and, incidentally, we also prove new fractional Poincaré-Wirtinger-type estimates. These inequalities display sharp limiting behaviours with respect to the fractional order of differentiability. As a byproduct, we obtain a new criterion for the embedding of the homogeneous Sobolev space in , valid in the subcritical regime and for . Our results are new even for the first eigenvalue of the fractional Laplacian and contain an optimal characterization for the positivity of the fractional Cheeger's constant.

Paper Structure

This paper contains 18 sections, 26 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Overview and goal of the paper
  3. Main Theorems
  4. Strategy of the proofs
  5. Plan of the paper
  6. Preliminaries
  7. Notation and functional spaces
  8. Two expedient lemmas
  9. Basic properties of fractional capacity
  10. Poincaré-type estimates on balls
  11. A Cheeger-type constant
  12. A Poincaré-type constant
  13. Poincaré inequalities
  14. Poincaré-Sobolev inequalities
  15. Proof of the Main Theorems
  16. ...and 3 more sections

Key Result

Theorem 1.4

Let $1 \le p < \infty$ and $0<s<1$ be such that $s\,p \le N$ and let $p \le q < p^*_s$. Let $0 < \gamma < 1$ and let $\Omega \subseteq \mathbb{R}^N$ be an open set. There exists a constant $\sigma = \sigma\left(N, p, s, q\right) > 0$ such that Moreover, for $1 \le p < \infty$ we have and for $1 \le p \le N$ we have

Theorems & Definitions (65)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Lower bound
  • Theorem 1.5: Upper bound
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10: Limiting cases
  • ...and 55 more