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Steklov-type 1D inequalities (a survey)

Alexander I. Nazarov, Alexandra P. Shcheglova

TL;DR

The survey consolidates sharp constants and symmetry properties for $1$-D Steklov-type inequalities across extensions, higher-order variants, periodic/non-homogeneous forms, and magnetic-term settings. It derives explicit optimal constants such as \\lambda_1(p,q)$, \\lambda_2(p,q,r)$, \\lambda_3(n,k,p,q)$, and \\lambda_4(\\alpha,m,q)$, and elucidates when extremals are symmetric or exhibit symmetry breaking, often reducing to constructions from base extremals $U_{p,q}$ and $V_{p,q}$. The work also highlights the interplay between endpoint conditions, periodicity, and higher-order derivatives, presenting both general principles (parity-driven symmetry, reductions to lower-order cases) and concrete formulas for central parameter regimes (notably $p=2$ and $q=\\infty$). The Appendix provides a rigorous symmetry proof for a key high-order, infinite-norm case, illustrating methodological approaches for identifying symmetric extremals. Overall, the paper advances understanding of when and how symmetry persists in sharp-constant inequalities and equips researchers with explicit extremals and criteria across a broad spectrum of 1D problems.

Abstract

We give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in $1$-dimensional functional inequalities.

Steklov-type 1D inequalities (a survey)

TL;DR

The survey consolidates sharp constants and symmetry properties for -D Steklov-type inequalities across extensions, higher-order variants, periodic/non-homogeneous forms, and magnetic-term settings. It derives explicit optimal constants such as \\lambda_1(p,q), \\lambda_3(n,k,p,q), and elucidates when extremals are symmetric or exhibit symmetry breaking, often reducing to constructions from base extremals and . The work also highlights the interplay between endpoint conditions, periodicity, and higher-order derivatives, presenting both general principles (parity-driven symmetry, reductions to lower-order cases) and concrete formulas for central parameter regimes (notably and ). The Appendix provides a rigorous symmetry proof for a key high-order, infinite-norm case, illustrating methodological approaches for identifying symmetric extremals. Overall, the paper advances understanding of when and how symmetry persists in sharp-constant inequalities and equips researchers with explicit extremals and criteria across a broad spectrum of 1D problems.

Abstract

We give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in -dimensional functional inequalities.

Paper Structure

This paper contains 6 sections, 12 theorems, 34 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A simplest extension of the inequality (\ref{['eq:Steklov2']})
  3. An extension of the inequality (\ref{['eq:Steklov1']})
  4. A higher-order extension of inequality (\ref{['eq:Steklov2']})
  5. A non-homogeneous inequality for periodic functions
  6. Appendix

Key Result

Theorem 1

The sharp constant in (eq:Friedrichs) is given by where $\mathfrak F (s) = \frac{\Gamma (s+1)}{s^s}$. The corresponding extremal function $U_{p,q}$ can be expressed in quadratures, does not change sign and is even with respect to $x=\frac{1}{2}$, see Fig. Fig2.

Figures (8)

  • Figure 1: The graphs of extremal functions for the inequalities (\ref{['eq:Steklov1']}) (thin line) and (\ref{['eq:Steklov2']}) (bold line).
  • Figure 2: The graphs of extremal functions in (\ref{['eq:Friedrichs']}) for $p=2$: $q=1$ (thin line) and $q=\infty$ (bold line).
  • Figure 3: The graphs of extremal functions in (\ref{['eq:Poincare']}) for $p=2$, $r=2$: $q=1$ (thin line) and $q=\infty$ (bold line).
  • Figure 4: The graphs of extremal functions in (\ref{['eq:high']}) for $p=2$, $q=2$: $n=4$, $k=2$ (thin line) and $n=3$, $k=1$ (bold line).
  • Figure 5: The graphs of extremal functions in (\ref{['eq:high']}) for $p=2$, $q=\infty$: $n=5$, $k=2$ (left) and $n=5$, $k=3$ (right).
  • ...and 3 more figures

Theorems & Definitions (25)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 3: BKN; see also Stech
  • Remark 5
  • Remark 6
  • Conjecture 1
  • ...and 15 more