Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-27
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FALSE: the regularized Perron envelope always attains the prescribed boundary data

Statement refuted

For every bounded plane domain and every continuous boundary datum, the regularized Perron envelope always attains the prescribed boundary values at every boundary point.

Facts & Assumptions

Given: The universal claim in the Statement refuted.

[L1]

For the punctured-disc datum 0 on ∣z∣=1 and 1 at the puncture, the annulus comparison in the published counterexample proves directly that every Perron lower function is at most 0 and the constant 0 belongs to the lower family (The punctured disc has an irregular boundary point and a continuous boundary datum with no harmonic solution).

Refutation

technique · direct
1.1L1

By [L1], the Perron envelope and its upper-semicontinuous regularization for this datum are identically 0.

2.1step 1.1given∎

Their limit at the puncture is therefore 0, while the prescribed value there is 1. This single bounded-domain datum refutes the universal boundary-attainment claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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