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Divergence right inverse on bounded connected Lipschitz domains

Statement

Let $D \subset R^{d}$ be nonempty, bounded, connected and Lipschitz, with $d \in {2, 3}$ . Under Countable Choice there is a bounded linear map $B_{D} : L_{0}^{2} (D) \to H_{0}^{1} (D; R^{d})$ such that $div B_{D} q = q, ∥ \nabla B_{D} q ∥_{2} \leq C_{d i v} (D) ∥ q ∥_{2} .$ For a chosen finite cover and tree decomposition from Finite star-shaped cover of a bounded Lipschitz domain and Zero-mean L2 decomposition over a finite cover, one valid bound is $C_{d i v} (D) = \sum_{i} C_{U_{i}} D_{i}$ ; it is finite and independent of any finite-element mesh.

Facts & Assumptions

Given: Countable Choice; a bounded connected Lipschitz domain, the finite star-shaped cover, its zero-mean decomposition constants, and the local Bogovskii right inverses.

[A1]

Countable Choice is inherited through the local L2 divergence right-inverse extension. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The cover has finitely many star-shaped $U_{i}$ and every $q \in L_{0}^{2} (D)$ decomposes linearly into $q_{i} \in L_{0}^{2} (U_{i})$ with $q = \sum_{i} q_{i}$ and $∥ q_{i} ∥_{2} \leq D_{i} ∥ q ∥_{2}$ . (Finite star-shaped cover of a bounded Lipschitz domain, Zero-mean L2 decomposition over a finite cover)

[F2]

Each local operator maps $L_{0}^{2} (U_{i})$ boundedly into $H_{0}^{1} (U_{i})^{d}$ with divergence $q_{i}$ and gradient bound $C_{U_{i}}$ . (Bounded Bogovskii right inverse on a star-shaped domain)

[F3]

Zero-boundary fields on bounded open sets have zero-mean divergence. (Zero mean of the divergence of a zero-boundary field)

Proof

1.1A1F1F2construct

For $q \in L_{0}^{2} (D)$ , form $q_{i}$ by [F1], apply the local theorem [F2], extend each $B_{U_{i}} q_{i}$ by zero to $D$ , and define $B_{D} q = \sum_{i} \tilde{B_{U_{i}} q_{i}}$ . The extension belongs to $H_{0}^{1} (D)^{d}$ : approximate in $H^{1} (U_{i})$ by compactly supported smooth fields and extend those by zero.

2.1A1F1F2step 1.1

Zero extension preserves weak derivatives because the approximating functions are compactly supported inside $U_{i}$ ; hence $div \tilde{B_{U_{i}} q_{i}}$ is the zero extension of $q_{i}$ . Summing and using [F1] gives $div B_{D} q = q$ . Linearity follows from the linear decomposition and local operators.

2.2A1F1F2algebrastep 1.1

The triangle inequality and [F1]–[F2] yield $∥ \nabla B_{D} q ∥_{2} \leq \sum_{i} C_{U_{i}} ∥ q_{i} ∥_{2} \leq (\sum_{i} C_{U_{i}} D_{i}) ∥ q ∥_{2}$ . Each sum is finite, and its ingredients depend only on the chosen domain cover, not on a mesh parameter.

3.1A1F3step 2.1∎

Conversely, any $v \in H_{0}^{1} (D)^{d}$ has zero-mean divergence by [F3], so $L_{0}^{2} (D)$ is the compatible pressure range. The identity in step 2.1 proves that every such datum is attained by the constructed right inverse.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapters 53–54 (complete course notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)