CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Locally integrable weakly harmonic functions are smooth

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . If $f \in L_{l o c}^{1} (Ω)$ and $Δ T_{f} = 0$ , then $f = h$ almost everywhere for a unique smooth harmonic $h$ .

Proof

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $f \in L_{l o c}^{1} (Ω)$ and $Δ T_{f} = 0$ .

1.1

Weyl's lemma for the Laplacian supplies a smooth harmonic $h$ with $T_{f} = T_{h}$ [given].

2.1

Equality of regular distributions implies $f = h$ almost everywhere, and uniqueness is inherited [step 1.1]. ∎

Depends on

Dependency tree · 2 levels

Sources

PDE source treatment