|
|
|
|
Perron family
|
(Definition)
|
|
Definition 1 Let $G \subset {\mathbb{C}}$ be a region, $\partial_\infty G$ the extended boundary of $G$ and $S(G)$ the set of subharmonic functions on $G$ then if $f \colon \partial_\infty G \to {\mathbb{R}}$ is a continuous function then the set \begin{equation*} {\mathcal{P}}(f,G) := \{ \varphi : \varphi \in S(G) \text{ and } \limsup_{z \to a} \varphi(z) \leq f(a) \text{ for all $a \in
\partial_\infty G$} \} , \end{equation*}is called the Perron family.
One thing to note is the ${\mathcal{P}}(f,G)$ is never empty. This is because $f$ is continuous on $\partial_\infty G$ it attains a maximum, say $|f| < M$ then the function $\varphi(z) := -M$ is in ${\mathcal{P}}(f,G)$
Definition 2 Let $G \subset {\mathbb{C}}$ be a region and $f \colon \partial_\infty G \to {\mathbb{R}}$ be a continuous function then the function $u \colon G \to {\mathbb{R}}$ \begin{equation*} u(z) := \sup \{ \phi : \phi \in {\mathcal{P}}(f,G) \} , \end{equation*}is called the Perron function associated with $f$
Here is the reason for all these definitions.
Theorem 1 Let $G \subset {\mathbb{C}}$ be a region and suppose $f \colon \partial_\infty G \to {\mathbb{R}}$ is a continuous function. If $u \colon G \to {\mathbb{R}}$ is the Perron function associated with $f$ then $u$ is a harmonic function.
Compare this with Rado's theorem which works with harmonic functions with range in ${\mathbb{R}}^2$ but also gives a much stronger statement.
- 1
- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
|
"Perron family" is owned by jirka.
|
|
(view preamble | get metadata)
Cross-references: stronger, range, harmonic function, definitions, function, continuous function, subharmonic functions, extended boundary, region
This is version 3 of Perron family, born on 2004-04-22, modified 2005-03-07.
Object id is 5797, canonical name is PerronFamily.
Accessed 2388 times total.
Classification:
| AMS MSC: | 31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
