ROBIN FUNCTIONS AND ENERGY FUNCTIONALS OF MULTIPLY CONNECTED DOMAINS

The Robin function of a planar domain is a generalization of Green's function. It can be used to represent the solutions of mixed boundary-value problems for harmonic functions. Here it is combined with a variational method to solve certain extremal problems for the energy functional of a multiply connected domain. Some deeper properties of the Robin function are then explored. An allied system of conformal invariants called the Robin matrix is introduced and is compared with the classical Riemann matrix of a finitely connected domain.