CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A gradient-only Poincare estimate needs normalisation

Statement refuted

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 2$ and $1 < p < n$ with $p^{*} = \frac{n p}{n - p}$ , and let $Ω$ be a nonempty bounded connected domain with smooth boundary. The following two assertions are false:

there is a finite constant $C$ with $∥ u ∥_{L^{p^{*}} (Ω)} \leq C ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W^{1, p} (Ω; K)$ ;
there is a finite constant $C$ with $∥ u ∥_{L^{p} (Ω)} \leq C ∥ D u ∥_{L^{p} (Ω)}$ for every $u \in W^{1, p} (Ω; K)$ .

A condition excluding nonzero constants is necessary; mean subtraction, zero trace, and vanishing on a set of positive measure are standard normalisations.

Facts & Assumptions

Given: Countable Choice; $n \geq 2$ ; $1 < p < n$ ; the nonempty bounded connected smooth domain $Ω_{0} = Ω$ of the statement; and a test function $φ \in C_{c}^{\infty} (Ω_{0})$ .

[F1]

$W^{1, p} (Ω_{0})$ consists of the $L^{p}$ classes whose first weak derivatives exist as $L^{p}$ classes; the weak derivative $D_{i} u$ is characterized by $\int_{Ω_{0}} u \partial_{i} φ = - \int_{Ω_{0}} D_{i} u φ$ for every test function (Integer-order Sobolev spaces and their norms, Weak derivative of a locally integrable function).

[F2]

$L^{p}$ consists of almost-everywhere classes, and the constant class $u \equiv 1$ lies in $L^{p} (Ω_{0})$ because $Ω_{0}$ contains a ball and is bounded, so it has finite positive measure (The space $L^{p} (μ)$ as the quotient by null functions, Euclidean balls have positive finite Lebesgue measure, Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F3]

On sigma-finite products nonnegative measurable functions may be integrated in either order (Tonelli-Fubini) (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F4]

If $G$ is differentiable everywhere on a compact interval and $G^{'}$ is integrable, then $\int_{a}^{b} G^{'} = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ). Applied to a compactly supported smooth real section, or separately to both components of a complex section, its derivative integral is zero.

[F5]

The Sobolev conjugate satisfies $p^{*} > p$ and $\frac{1}{p^{*}} = \frac{1}{p} - \frac{1}{n}$ , so the two exponents in the refuted assertions are distinct and both finite (The Sobolev conjugate exponent and the scaling identity).

Counterexample

technique · direct

1.1F1F2F3F4givenalgebra

The constant function is a Sobolev function with zero gradient. Take $u \equiv 1$ on $Ω_{0}$ . By [F2], $u \in L^{p} (Ω_{0}; K)$ (the class is represented by the constant function; for $K = C$ it is the complex constant). Fix $i$ and let $φ \in C_{c}^{\infty} (Ω_{0})$ ; extend $φ$ by zero to $R^{n}$ . Writing ${\hat{x}}_{i}$ for the coordinates other than $x_{i}$ and using Fubini [F3], $\int_{Ω_{0}} \partial_{i} φ = \int_{R^{n}} \partial_{i} φ = \int_{R^{n - 1}} (\int_{R} \partial_{i} φ (\dots, t, \dots) d t) d {\hat{x}}_{i} = 0$ , because for each fixed ${\hat{x}}_{i}$ the inner function $t \mapsto φ (\dots, t, \dots)$ is compactly supported and smooth, so [F4] gives vanishing integral. Hence $\int_{Ω_{0}} u \partial_{i} φ = 0 = \int_{Ω_{0}} 0 \cdot φ$ for every test function, i.e. the zero class is the weak derivative $D_{i} u$ by [F1]; in particular $D u = 0$ as an element of $L^{p} (Ω_{0}; K)$ .

2.1F2F5step 1.1givenalgebra∎

Both proposed inequalities fail. Since $u \equiv 1$ and $D u = 0$ , the two sides of the first proposed estimate are $∥ u ∥_{L^{p^{*}} (Ω_{0})} = ∣ Ω_{0} ∣^{1 / p^{*}} > 0$ by [F2] and $∥ D u ∥_{L^{p} (Ω_{0})} = 0$ ; the second has $∥ u ∥_{L^{p} (Ω_{0})} = ∣ Ω_{0} ∣^{1 / p} > 0$ and again $∥ D u ∥_{L^{p} (Ω_{0})} = 0$ . No finite $C$ can satisfy $∣ Ω_{0} ∣^{1 / p^{*}} \leq C \cdot 0$ or $∣ Ω_{0} ∣^{1 / p} \leq C \cdot 0$ . Finally $u - u_{Ω_{0}} = 0$ and $u$ does not vanish on any set of positive measure and satisfies no vanishing trace condition, so mean subtraction annihilates precisely this witness while trace or positive-measure zero-set normalisations exclude it; a condition excluding nonzero constants is therefore necessary.

Source notes

The refutation is the explicit constant-function witness against the un-normalised inequalities. Kinnunen's Theorem 3.47, printed pp. 90-91, states the mean-zero form; the constant witness directly shows why that normalisation matters. The computation of the weak gradient of a constant uses only Fubini and the one-dimensional fundamental theorem, so it applies to every nonempty open $Ω_{0}$ of finite measure, not only to a ball; the same computation proves the claim on the arbitrary finite-measure domain in the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

67 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juha Kinnunen, Sobolev Spaces (standard reference, not scraped)