Dirichlet principle for the Royden compactification
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 \textit{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.
scientific article; zbMATH DE number 4042265