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The mixed pressure has the mean-zero gauge

Statement

Assume Countable Choice. Let $D \subset R^{d}$ be bounded and open, let $0 < λ < \infty$ , and let $u \in H_{0}^{1} (D; R^{d})$ . Since $D$ has finite measure, Hölder's inequality makes every $L^{2} (D)$ representative integrable, and almost-everywhere equal representatives have the same integral. Thus the local mean-zero space $Q_{D} : = {q \in L^{2} (D; R) : \int_{D} q = 0} .$ Set $p = - λ div u$ . Then $p \in Q_{D}$ . If additionally $d \in {2, 3}$ and $D$ is nonempty, connected, and Lipschitz, this $Q_{D}$ is exactly the space $Q = L_{0}^{2} (D)$ of Stokes velocity and pressure spaces and weak forms, so $p$ belongs to the mixed elasticity model's pressure space.

Facts & Assumptions

Given: Countable Choice, a bounded open $D$ , a finite $λ > 0$ , $u \in H_{0}^{1} (D; R^{d})$ , the local pressure space $Q_{D}$ in the Statement, and $p = - λ div u$ .

[A1]

Countable Choice is required by the zero-mean-divergence supplier. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Every zero-boundary field has an $L^{2}$ divergence with integral zero. (Zero mean of the divergence of a zero-boundary field)

[F2]

On bounded $D$ , every $L^{2} (D)$ representative is integrable by Hölder, the integral is unchanged under almost-everywhere equality, and it is linear; hence $Q_{D} = {q \in L^{2} (D; R) : \int_{D} q = 0}$ is a well-defined linear subspace. When $d \in {2, 3}$ and $D$ is nonempty, bounded, connected, and Lipschitz, the imported Stokes definition uses exactly this $Q_{D}$ . (Holder's inequality for integrals, including the endpoint cases, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree, The Lebesgue integral is linear on $L^{1} (μ)$ , Stokes velocity and pressure spaces and weak forms)

Proof

1.1A1F1F2given

Under [A1], [F1] gives $div u \in L^{2} (D)$ and $\int_{D} div u = 0$ . By the definition of $Q_{D}$ in [F2], this means $div u \in Q_{D}$ .

2.1givenF2step 1.1algebra∎

Multiplication by the finite scalar $- λ$ preserves membership in the linear subspace $Q_{D}$ . Therefore $p = - λ div u \in Q_{D}$ . In the stated connected Lipschitz model domain, [F2] identifies $Q_{D}$ with $Q$ , giving the model-space conclusion as well.

Depends on

Used by

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Sources

Long Chen, Introduction to Linear Elasticity and Its Finite Element Approximation (complete 13-page notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete 691-page text) (standard reference, not scraped)