Robin functions and energy functionals of multiply connected domains
The Robin function of a finitely connected domain \(\Omega\) is the function R(z,\(\zeta)\) harmonic in \(\Omega\) except for a logarithmic singularity at \(\zeta\), with \(R=0\) on one set of boundary components and \(\partial R/\partial n=0\) on the others. The Robin function generalizes Green's function and can be used similarly to solve mixed boundary-value problems for harmonic functions. The existence of the Robin function is proved by displaying it as a solution to a certain extremal problem involving transfinite diameter. The Robin function generates a Robin matrix analogous to the classical Riemann matrix of periods of harmonic conjugates of harmonic measures of boundary components. Some comparison theorems for the Robin function and the Robin matrix are established. Finally, the Robin matrix is used together with a variational method to solve an extremal problem for the energy functional of \(\Omega\). The energy functional arose and was given a physical interpretation in the authors' previous study of conformal mappings onto nonoverlapping regions [Ann. Acad. Sci. Fenn., Ser. A I, to appear].
scientific article; zbMATH DE number 4116883