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On the best constant in {G}affney inequality

Gyula Csato, Bernard Dacorogna, Swarnendu Sil

TL;DR

This work identifies the best constant in Gaffney's inequality for differential forms and links its vanishing to geometric boundary convexity. Using boundary-operator expressions $L^{ν}$ and $K^{ν}$ tied to principal curvatures, it proves that $C_T(Ω,k)=1$ is equivalent to the domain being $(n-k)$-convex, the sharper inequality, non-attainment of the supremum, and scale invariance, with a dual, analogous result for $C_N$. The authors develop algebraic and geometric tools to express boundary contributions purely in terms of boundary geometry, and they show that in general non-convex domains the constants can be arbitrarily large, while for polytopes the inequality is exact with constant 1. The results extend to piecewise-smooth and polyhedral domains via integral identities and curvature sums, linking spectral-type inequalities to boundary geometry and providing sharp Korn-type corollaries in special cases.

Abstract

We discuss the value of the best constant in Gaffney inequality namely $$ \lVert \nabla ω\rVert_{L^{2}}^{2}\leq C\left( \lVert dω\rVert_{L^{2}}^{2}+\lVert δω\rVert_{L^{2}% }^{2}+\lVert ω\rVert_{L^{2}}^{2}\right) $$ when either $ν\wedgeω=0$ or $ν\,\lrcorner\,ω=0$ on $\partialΩ.$

On the best constant in {G}affney inequality

TL;DR

This work identifies the best constant in Gaffney's inequality for differential forms and links its vanishing to geometric boundary convexity. Using boundary-operator expressions and tied to principal curvatures, it proves that is equivalent to the domain being -convex, the sharper inequality, non-attainment of the supremum, and scale invariance, with a dual, analogous result for . The authors develop algebraic and geometric tools to express boundary contributions purely in terms of boundary geometry, and they show that in general non-convex domains the constants can be arbitrarily large, while for polytopes the inequality is exact with constant 1. The results extend to piecewise-smooth and polyhedral domains via integral identities and curvature sums, linking spectral-type inequalities to boundary geometry and providing sharp Korn-type corollaries in special cases.

Abstract

We discuss the value of the best constant in Gaffney inequality namely when either or on

Paper Structure

This paper contains 12 sections, 20 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Some generalities
  4. The main theorem
  5. Statement of the theorem
  6. Some algebraic results
  7. Calculation of sums of principal curvatures
  8. Formulas for $L^{\nu}$ and $K^{\nu}$ in terms of principal curvatures
  9. Proof of the main theorem
  10. Some examples
  11. The case of polytopes
  12. Appendix: an integral identity for differential forms

Key Result

Theorem 1

Let $0\leq k\leq n$ and $\Omega\subset\mathbb{R}^{n}$ be a bounded open $C^{2}$ set. Then there exists a constant $C=C\left( \Omega,k\right) >0$ such that for every $\omega\in W_{T}^{1,2}\left( \Omega;\Lambda^{k}\right) \cup W_{N}^{1,2}\left( \Omega;\Lambda^{k}\right) .$

Theorems & Definitions (36)

  • Theorem 1: Gaffney inequality
  • Proposition 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Lemma 10
  • ...and 26 more