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An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

Francesca Bianchi, Lorenzo Brasco

Abstract

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman and Taylor, in the limit as $s$ goes to $1$. The limit as $s$ goes to $1/2$ is carefully analyzed, as well.

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

Abstract

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman and Taylor, in the limit as goes to . The limit as goes to is carefully analyzed, as well.
Paper Structure (21 sections, 16 theorems, 252 equations, 6 figures)

This paper contains 21 sections, 16 theorems, 252 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Goal of the paper
  3. The Croke-Osserman-Taylor inequality
  4. Main results
  5. Comments on the proofs
  6. Plan of the paper
  7. Preliminaries
  8. Notation
  9. Fatness of the complement of a multiply connected set
  10. Functional spaces
  11. Some facts from the theory of fractional Sobolev spaces
  12. Basics of fractional capacity
  13. A Maz'ya--type Poincaré inequality
  14. A geometric lower bound in the plane
  15. Proof of Theorem \ref{['teo:main']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $1/2<s<1$, there exists a constant $\vartheta_s>0$ such that for every $\Omega\subseteq\mathbb{R}^2$ open multiply connected set of order $k\in\mathbb{N}\setminus\{0\}$, we have Moreover, the constant $\vartheta_s$ has the following asymptotic behaviours

Figures (6)

  • Figure 1: The construction of disks and squares in the proof of Lemma \ref{['lm:rettangolo distante']}, for the cases $k=1$, $k=2$ or $k=3$ (i. e. $\delta=2$). Each disk contains at least a point belonging to $\mathbb{R}^2\setminus \Omega$. The reliable squares are those for which such a point can be "connected" to the boundary of the "cell" containing it, with a continuum lying outside of $\Omega$.
  • Figure 2: A zooming on a reliable square $Q_{5/2}(P_{j,m})$. The bold line corresponds to a continuum which connects the point $X_{j,m}$ to the boundary of the "cell", lying outside of $\Omega$.
  • Figure 3: The two quantities $R_\omega(x)$ and $r_\omega(x)$.
  • Figure 4: The geometric configuration of Lemma \ref{['lm:poincare_cap']}: we have a smooth function defined on the square, which vanishes on the dashed neighborhood of the vertical line (i.e. the set $\Sigma$). The relative fractional capacity of $\Sigma$ is computed with respect to the surronding disk.
  • Figure 5: The set $\Omega_k$ of Theorem \ref{['teo:optimal']} point (2), for $k=25$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Definition
  • Theorem 1.1: Main Theorem
  • Theorem 1.2: Optimality
  • Definition
  • Lemma 2.1: Taylor's fatness lemma
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 25 more