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The Perron envelope and its regularization

Definition

Let $Ω$ be a bounded complex domain and let $φ : \partial Ω \to R$ be continuous. The Perron envelope is the pointwise supremum $U_{φ} (z) : = \sup {v (z) : v \in P (φ, Ω)}, z \in Ω .$

Its regularized Perron envelope is defined directly by $H_{φ} (z) : = \lim_{ρ ↓ 0} \sup {U_{φ} (w) : w \in Ω, ∣ w - z ∣ < ρ} .$

Remarks

This limsup is meaningful with values in $[- \infty, + \infty]$ before any boundedness theorem is used. The next lemma proves $\min_{\partial Ω} φ \leq U_{φ} \leq \max_{\partial Ω} φ$ , so in the present Perron setting $H_{φ}$ is finite-valued and is exactly the upper-semicontinuous regularization $U_{φ}^{*}$ .

Depends on

The Perron lower family for continuous boundary data

Used by

Barriers and regular boundary points Definition
The regularized Perron envelope is harmonic Theorem

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Sources

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