ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

The essential supremum precedes the Holder representative in De Giorgi theory

Example

Example. On $Ω = B_{1} (0) \subset R^{2}$ let $u$ be the zero class of $H^{1} (Ω)$ (the class of the function that vanishes a.e.), and let $\hat{u} = 1_{{0}}$ be the representative that equals $1$ at the origin and $0$ elsewhere. Then:

$u$ is a weak solution of $- Δ u = 0$ on $Ω$ (Local weak solutions of a divergence-form operator);
$\hat{u}$ differs from the zero function on the Lebesgue-null set ${0}$ , so $u$ and $\hat{u}$ determine the same class and the same weak derivatives (Weak differentiation ignores null-set changes);
$\sup_{Ω} \hat{u} = 1$ while ${ess sup}_{Ω} u = 0$ (The essential supremum of a measurable function with respect to a measure), so the pointwise supremum of an arbitrary representative is not the quantity controlled by the local boundedness estimate De Giorgi local boundedness of homogeneous subsolutions or by the Harnack bound Harnack inequality for nonnegative weak solutions. To read these class estimates as pointwise bounds, use the continuous representative produced by De Giorgi-Nash interior Holder regularity for divergence-form equations; its pointwise and essential extrema agree.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; the unit disc $Ω = B_{1} (0) \subset R^{2}$ ; the zero class $u \in H^{1} (Ω)$ and the representative $\hat{u} = 1_{{0}}$ .

[F1]

The local weak formulation: $u$ is a local weak solution of $- Δ u = 0$ on $Ω$ if $\int_{Ω} \nabla u \cdot \nabla v d x = 0$ for every $v \in H_{0}^{1} (Ω)$ ; the zero class satisfies this identically (Local weak solutions of a divergence-form operator).

[F2]

Weak derivatives depend only on the class: two $L_{l o c}^{1}$ representatives of the same class have the same weak derivatives, and the set ${0}$ is Lebesgue-null, so $\hat{u}$ and the zero function determine the same class (Weak differentiation ignores null-set changes, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Essential versus pointwise suprema: the essential supremum of a class is the infimum of the essential bounds, hence ${ess sup}_{Ω} u = 0$ for the zero class, whereas the pointwise supremum of the particular function $\hat{u}$ is $\sup_{Ω} \hat{u} = 1$ (The essential supremum of a measurable function with respect to a measure, Local Hölder and scaled C-two-alpha norms on balls).

[F4]

The estimates of the page are stated for essential extrema of classes: the local boundedness theorem bounds ${ess sup}_{B_{ρ R}} u$ by an $L^{p}$ mean of the class, and the Harnack inequality bounds ${ess sup}_{B_{R / 2}} u$ by ${ess inf}_{B_{R / 2}} u$ (De Giorgi local boundedness of homogeneous subsolutions, Harnack inequality for nonnegative weak solutions, De Giorgi-Nash interior Holder regularity for divergence-form equations).

Verification

1.1givenF1

The zero class is a weak solution. For every $v \in H_{0}^{1} (Ω)$ one has $\int_{Ω} \nabla u \cdot \nabla v d x = 0$ because $\nabla u = 0$ a.e. for the zero class, so [F1] exhibits $u$ as a local weak solution of $- Δ u = 0$ on $Ω$ ; equivalently, the classical zero solution restricted to $Ω$ .

2.1step 1.1F2

The two representatives differ on a null set. The set ${0}$ has Lebesgue measure zero, so $\hat{u} = 0$ a.e. and $\hat{u}$ represents the class $u$ ; by [F2] $\hat{u}$ and the zero function have the same weak derivatives, so every weak formulation tested against $\hat{u}$ gives the same value as against the zero function.

3.1step 2.1F3F4∎

The suprema differ, so only the essential supremum is controlled. By [F3], ${ess sup}_{Ω} u = 0$ while $\sup_{Ω} \hat{u} = 1$ : the pointwise supremum of the particular representative $\hat{u}$ exceeds the essential supremum of the class. The local boundedness and Harnack estimates of [F4] control only essential extrema of the class, so they cannot be applied to an arbitrary pointwise representative; the class estimates give pointwise bounds for the Holder representative produced by De Giorgi-Nash interior Holder regularity for divergence-form equations, which for the zero class is the zero function and for which pointwise and essential extrema agree. All verifications use the explicit functions and the cited interface items, with no choice principle beyond the declared Axiom of Choice and Countable Choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

58 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

Brian Krummel, DeGiorgi-Nash lecture notes (15 March 2016; complete 9-page notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete author scan, 118 sheets reproducing the 223 printed pages of the manuscript, two logical pages per sheet) (standard reference, not scraped)