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The Perron lower family for continuous boundary data

Definition

Let $Ω \subseteq C$ be a bounded complex domain and let $φ : \partial Ω \to R$ be continuous. A function $v : Ω \to [- \infty, \infty)$ is a Perron lower function for $(Ω, φ)$ when:

$v$ is subharmonic on $Ω$ ;
for every $ζ \in \partial Ω$ , ${lim sup}_{\begin{array}{c} z \to ζ \\ z \in Ω \end{array}} v (z) \leq φ (ζ) .$

The collection of all such functions is the Perron lower family and is denoted $P (φ, Ω)$ .

Remarks

The word lower refers to the boundary condition: members of the family are subharmonic functions that stay below the prescribed boundary datum in the limsup sense.

Depends on

Subharmonic functions on plane domains

Used by

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Sources

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