DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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Barriers and regular boundary points

Definition

Let $Ω \subseteq C$ be a bounded complex domain and let $ζ \in \partial Ω$ .

A barrier at $ζ$ is a subharmonic function $b : Ω \to [- \infty, 0)$ such that:

$b (z) \to 0$ as $z \to ζ$ with $z \in Ω$ ;
for every neighbourhood $V$ of $ζ$ there is a constant $c_{V} < 0$ with ${lim sup}_{\begin{array}{c} z \to η \\ z \in Ω \end{array}} b (z) \leq c_{V} (η \in \partial Ω ∖ V) .$

The boundary point $ζ$ is regular when for every continuous boundary datum $φ : \partial Ω \to R$ , the regularized Perron envelope $H_{φ}$ satisfies $\lim_{\begin{array}{c} z \to ζ \\ z \in Ω \end{array}} H_{φ} (z) = φ (ζ) .$

Remarks

The barrier is global on $Ω$ , but the second clause is exactly what makes a local peak function sufficient: once one is globalized, it is automatically separated from $0$ away from the marked boundary point.

Depends on

Used by

Dependency tree · two levels

3 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

Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, 2nd ed. (standard reference, not scraped)