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Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond

Lorenzo Brasco, Luca Briani, Francesca Prinari

TL;DR

The paper addresses the existence of extremals for the sharp Poincaré-Sobolev embedding constant $\lambda_p,q(\Omega)$ on unbounded periodically perforated domains $\Omega$ generated by a hole $K$ with positive $p$-capacity $\mathrm{cap}_p(K;Q_1)$. It develops a Lieb-type variational scheme with a vanishing confinement term and a Maz'ya-Lieb-type capacity estimate to obtain compactness up to translations and to extract a nontrivial minimizer that attains the constant, for all $1<p<\infty$ and $q>p$, including the endpoint $q=\infty$. The main results prove existence of extremals for $\Omega$ that are periodic in all directions (and extend to sets periodic in some directions and bounded in others), under the sharp capacitary condition on $K$, with an explicit bound that the extremal attains its maximum on the non-hole periodic cell. The work also provides examples showing that breaking periodicity can either destroy or preserve existence, highlighting delicate global-geometric effects and offering a framework for spectral optimization in perforated media.

Abstract

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond

TL;DR

The paper addresses the existence of extremals for the sharp Poincaré-Sobolev embedding constant on unbounded periodically perforated domains generated by a hole with positive -capacity . It develops a Lieb-type variational scheme with a vanishing confinement term and a Maz'ya-Lieb-type capacity estimate to obtain compactness up to translations and to extract a nontrivial minimizer that attains the constant, for all and , including the endpoint . The main results prove existence of extremals for that are periodic in all directions (and extend to sets periodic in some directions and bounded in others), under the sharp capacitary condition on , with an explicit bound that the extremal attains its maximum on the non-hole periodic cell. The work also provides examples showing that breaking periodicity can either destroy or preserve existence, highlighting delicate global-geometric effects and offering a framework for spectral optimization in perforated media.

Abstract

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

Paper Structure

This paper contains 18 sections, 17 theorems, 278 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A bestiary of open sets
  3. Main results
  4. Comments on the proof
  5. Plan of the paper
  6. Preliminaries
  7. Notation
  8. Two technical results
  9. Periodically perforated sets
  10. A Maz'ya-Lieb--type estimate
  11. A compactness result
  12. A well-prepared minimizing sequence
  13. Existence of extremals: periodic sets
  14. Existence of extremals: more general sets
  15. Breaking the periodicity: examples
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $E\subseteq\mathbb{R}^N$ be an open set. Let $T:\mathbb{R}^N\to\mathbb{R}^N$ be an invertible affine map, i.e. there exist an invertible matrix $A\in\mathcal{M}_N(\mathbb{R})$ and a vector $\mathbf{b}\in\mathbb{R}^N$ such that Then for every $1\le p<\infty$ and $q\ge 1$ satisfying exponents we haveWhen $q=\infty$, we agree that $(q-p)/q=1$. where $L\ge \ell>0$ are the maximal and minimal eige

Figures (5)

  • Figure 1: The basic cube $Q_{1/2}$ containing a compact set $K$ (in bold line).
  • Figure 2: A periodically perforated open set $\Omega\subseteq\mathbb{R}^2$, generated by $K$. Here we took $t_1=2$, $t_2=1/2$, i.e. the set is obtained by gluing together translated copies of the cube in Figure \ref{['fig:cubo_base']}, which has been stretched horizontally by a factor $2$ and compressed vertically by a factor $1/2$.
  • Figure 3: In bold line, an example of a two-dimensional open set which is periodic in the first coordinate direction and bounded in the second one. We can extend it periodically in the vertical direction, by adding translated copies of the original set: the new set falls into the realm of Theorem \ref{['teo:maininfty']}. The rectangles in dashed line represent the periodicity cells.
  • Figure 4: The tubular neighborhood of a helical spring: by Theorem \ref{['teo:pupazzo']}, for this set we have existence of extremals for $\lambda_{p,q}$, with $q>p$.
  • Figure 5: The set $\Omega_R$ of Example \ref{['exa:perforato_exa']}: we remove from $\mathbb{R}^N$ a periodic array of equal balls, except for a larger one, centered at the origin for example. For this set $\lambda_{p,q}$ fails to have extremals.

Theorems & Definitions (44)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 34 more