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Poincare-Wirtinger fails on disconnected bounded domains

Statement refuted

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω = B (0, 1) \cup B (3 e_{1}, 1) \subset R^{2}$ (two disjoint unit balls) and $u = 1_{B (3 e_{1}, 1)}$ . Then $D u = 0$ almost everywhere and $u$ is not almost everywhere constant on $Ω$ , so $u - u_{Ω}$ is nonzero on a set of positive measure and $∥ u - u_{Ω} ∥_{L^{p} (Ω)} > 0$ : the mean-zero Poincare-Wirtinger inequality is false on disconnected bounded open sets, and connectedness is essential for the single-global-mean normalisation.

Facts & Assumptions

Given: Countable Choice; the open bounded set $Ω = B (0, 1) \cup B (3 e_{1}, 1) \subseteq R^{2}$ ; the indicator $u = 1_{B (3 e_{1}, 1)}$ ; and $1 \leq p < \infty$ .

[F1]

Every Euclidean ball has positive finite Lebesgue measure, measures are monotone and countably additive on disjoint measurable sets (Euclidean balls have positive finite Lebesgue measure, Measures are monotone, Measures on sigma-algebras).

[F2]

$D_{i} u$ is the weak derivative if $\int_{Ω} u \partial_{i} φ = - \int_{Ω} D_{i} u φ$ for every test function $φ$ , and constant classes have zero weak derivative (Weak derivative of a locally integrable function).

[F3]

The two open balls are disjoint because $∣ 3 e_{1} ∣ = 3 > 2$ ; their closures are compact, and each open ball has positive finite measure; the ball average is the normalized integral (For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact, The average of a locally integrable function over a Euclidean ball, The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

Countable Choice is assumed; classical smooth derivatives are weak derivatives (Classical derivatives agree with weak derivatives). Translated balls have equal measure (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F5]

With the additional Axiom of Choice (The Axiom of Choice), the mean-zero Poincare-Wirtinger inequality holds on bounded John domains in dimensions $n \geq 2$ for every $1 \leq p < \infty$ (The mean-zero Poincare inequality on bounded John domains). This positive comparison uses the stronger hypothesis; the two-ball counterexample needs only Countable Choice.

Counterexample

technique · direct

1.1F1F2F3F4givenalgebra

The function $u$ is locally constant, hence smooth on $Ω$ , with all classical partial derivatives zero. By [F4] its weak gradient is zero. Since $∣ u ∣ \leq 1$ and $Ω$ has finite measure, $u \in W^{1, p} (Ω)$ and $∥ D u ∥_{p} = 0$ . The two balls have equal measure by [F4], so $∣ Ω ∣ = 2 ∣ B (0, 1) ∣$ by [F1].

2.1F1F3step 1.1givenalgebra

The mean and the oscillation. Since $u = 0$ on $B (0, 1)$ and $u = 1$ on $B (3 e_{1}, 1)$ and the two balls have equal measure, $u_{Ω} = ∣ Ω ∣^{- 1} \int_{Ω} u = ∣ B (3 e_{1}, 1) ∣ / (2 ∣ B (0, 1) ∣) = 1 / 2$ . Therefore $∣ u - u_{Ω} ∣ = 1 / 2$ on both balls, and $∥ u - u_{Ω} ∥_{L^{p} (Ω)}^{p} = \int_{Ω} (1 / 2)^{p} = 2 ∣ B (0, 1) ∣ 2^{- p} > 0$ , while $u$ is not almost everywhere constant on $Ω$ (it takes the values $0$ and $1$ on sets of positive measure).

3.1F1F5step 1.1step 2.1givenalgebra∎

Failure and the role of connectedness. The mean-zero Poincare-Wirtinger inequality would require $∥ u - u_{Ω} ∥_{L^{p} (Ω)} \leq C ∥ D u ∥_{L^{p} (Ω)}$ for a constant $C$ depending only on the domain and $p$ ; but the left side is $2^{1 / p} ∣ B (0, 1) ∣^{1 / p} / 2 > 0$ by step 2.1 while the right side is $0$ by step 1.1, so no finite $C$ exists. A zero-set normalisation on only one component does not repair the inequality: this very $u$ vanishes on $B (0, 1)$ , a set of half the domain measure. Each component must be normalised separately, or connectedness imposed; with the additional Axiom of Choice, bounded John domains, which are connected, satisfy the inequality by [F5].

Source notes

The counterexample is the standard two-ball two-valued function, matching the "what eliminates constants" discussion in Kinnunen's Remark 3.11 and Hunter's Chapter 4: the mean of a nonzero mean-zero function is the only quantity that can fail, and on a disconnected domain a locally constant function need not be constant. The computation uses only that the two balls have equal positive measure and that the gradient of a locally constant class vanishes.

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Sources

Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)