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Dirichlet principle for the Royden compactification. (English) Zbl 0639.30042

Let R be a hyperbolic Riemann surface, and denote by \(\Delta\) its Royden harmonic boundary. For a compact subset K of \(\Delta\) there is the important notation of Dirichlet capacity, c(K), developed by the author and M. Nakai [Isr. J. Math. 55, 15-32 (1986; Zbl 0605.30045)]. Given a Dirichlet function f on R, in the sense of Constantinescu-Cornea, the author establishes the following generalization of the Dirichlet principle: For a compact set \(K\subseteq \Delta\) with \(c(K)>0\) and the family \[ {\mathcal F}_ f=\{\phi: \phi \text{ is a Dirichlet function, }\phi =f \text{ q.e. on } K\}, \] there exists a unique harmonic function \(h_ f\in {\mathcal F}_ f\) satisfying \[ D(h_ f)=\inf \{D(\phi): \phi \in {\mathcal F}_{\phi}\}. \] Here \(D(\phi)=\int_{R}d\phi \wedge *d\phi\), and “q.e.” means up to a set of Dirichlet capacity zero.
Reviewer: J.L.Schiff

MSC:

30F20 Classification theory of Riemann surfaces

Citations:

Zbl 0605.30045
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References:

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