Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The role of topology and capacity in some bounds for principal frequencies

Francesco Bozzola, Lorenzo Brasco

TL;DR

The paper provides sharp lower bounds for generalized principal frequencies $\lambda_{p,q}(\Omega)$ in terms of the inradius $r_{\Omega}$, covering two regimes: topology-driven planar domains and capacity-driven higher-dimensional sets with non-removable points. It extends classical Makai–Hayman–Taylor results to the full $p$-Laplacian Poincaré–Sobolev family, introducing punctured Poincaré constants and a Maz'ya–Poincaré inequality to handle endpoint regimes and interpolation. The authors derive explicit bounds in 2D that depend on the topology (order of connectivity $k$) and show asymptotic sharpness, together with Moser–Trudinger-type limits for the critical case $p=2$. They also connect spectral bounds to geometric constants via Cheeger’s constant, proving a Buser-type inequality in the plane for multiply connected domains and discussing optimality and open problems. Collectively, the results provide a cohesive framework linking inradius, topology, capacity, and geometric inequalities to govern the sharp constants in Poincaré–Sobolev embeddings across dimensions and regimes.

Abstract

We prove a lower bound on the sharp Poincaré-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet-Laplacian. We also consider some limit situations, like the sharp Moser-Trudinger constant and the Cheeger constant. As a a byproduct of our discussion, we also obtain a Buser--type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

The role of topology and capacity in some bounds for principal frequencies

TL;DR

The paper provides sharp lower bounds for generalized principal frequencies in terms of the inradius , covering two regimes: topology-driven planar domains and capacity-driven higher-dimensional sets with non-removable points. It extends classical Makai–Hayman–Taylor results to the full -Laplacian Poincaré–Sobolev family, introducing punctured Poincaré constants and a Maz'ya–Poincaré inequality to handle endpoint regimes and interpolation. The authors derive explicit bounds in 2D that depend on the topology (order of connectivity ) and show asymptotic sharpness, together with Moser–Trudinger-type limits for the critical case . They also connect spectral bounds to geometric constants via Cheeger’s constant, proving a Buser-type inequality in the plane for multiply connected domains and discussing optimality and open problems. Collectively, the results provide a cohesive framework linking inradius, topology, capacity, and geometric inequalities to govern the sharp constants in Poincaré–Sobolev embeddings across dimensions and regimes.

Abstract

We prove a lower bound on the sharp Poincaré-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet-Laplacian. We also consider some limit situations, like the sharp Moser-Trudinger constant and the Cheeger constant. As a a byproduct of our discussion, we also obtain a Buser--type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.
Paper Structure (19 sections, 21 theorems, 248 equations, 3 figures)

This paper contains 19 sections, 21 theorems, 248 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Description of the results
  4. A glimpse of Cheeger's constant
  5. Plan of the paper
  6. Preliminaries
  7. Notation
  8. An extension operator
  9. Poincaré-Wirtinger inequalities
  10. A Maz'ya-Poincaré type inequality
  11. The case $N=2$
  12. Three technical facts
  13. Inradius and principal frequencies
  14. Embeddings for homogeneous spaces
  15. The case $p > N$
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $x_0 \in \mathbb{R}^N$ and $r > 0$. There exists a linear extension operator such that, for every $1\le p\le \infty$, it maps $W^{1,p}(B_r(x_0))$ to $W^{1,p}_{\rm loc}(\mathbb{R}^N)$. Moreover, for every $u\in W^{1,p}(B_r(x_0))$ and every $R > r$, it holds

Figures (3)

  • Figure 1: The set $\Omega_k$ for $k=7$
  • Figure 2: The graph of the funnel--type function $u_k$.
  • Figure 3: The set $\widetilde{\mathcal{Q}}_{k}$ for $k=17$: it is made of the "internal" perforated squares in grey.

Theorems & Definitions (48)

  • Definition 1.1
  • Remark 1.2: Comparison with previous results I
  • Remark 1.3: Comparison with previous results II
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 38 more