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On divergence operators: Free space and vanishing charges

Thierry De Pauw

Abstract

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(Ω;\mathbb{R}^m)$ or $v \in C_0(Ω;\mathbb{R}^m)$, i.e. we characterize those $F$ for which a solution $v$ exists whose norm is bounded by an appropriate norm of $F$. We assume $Ω$ satisfies a Poincaré inequality or an extension property. We apply the general theory to give examples of admissible $F$ in each case.

On divergence operators: Free space and vanishing charges

Abstract

We use localized topologies to prove existence and optimal regularity results for the divergence equation in critical cases or , i.e. we characterize those for which a solution exists whose norm is bounded by an appropriate norm of . We assume satisfies a Poincaré inequality or an extension property. We apply the general theory to give examples of admissible in each case.
Paper Structure (23 sections, 16 theorems, 102 equations)

This paper contains 23 sections, 16 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Foreword
  3. The case m=1
  4. Free space and vanishing charges
  5. Non-invertibility
  6. Localized locally convex topologies
  7. The localized topology P-Lip
  8. The localized topology W-p-TV
  9. The localized topology M-1-TV
  10. Notation and preliminaries
  11. Consequences of integration by parts and the closed range theorem
  12. Lebesgue-summable vector fields and the space F(Rm)
  13. Continuous vector fields vanishing at infinity and the space SCH-0(Rm)
  14. The Lipschitz-free space F(Omega)
  15. The localized topology P-Lip
  16. ...and 8 more sections

Key Result

Theorem A

Both divergence operators occurring in the diagrams below are surjective.

Theorems & Definitions (29)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem 3.1.4
  • proof
  • Theorem 3.2.4
  • proof
  • ...and 19 more