Minimizing operators on subregions

Abstract

For constructing harmonic functions with a prescribed local behavior, an operator method on arbitrary Riemann surfaces was recently introduced by the author [1]. We showed the existence of a normal linear operator minimizing the Dirichlet integral and referred to other operators to be given later. In the present paper, a general class of minimizing operators will be introduced, including the above operator as a special case. In the existence proof, use will be made of the extremal method presented in [21. Let R be an arbitrary Riemann surface'and G a subregion, compact or not, of finite or infinite genus, relatively bounded by a finite set a of closed analytic Jordan curves. Let v be a real single-valued function on a, harmonic in an open set containing ax. A normal linear operator L in G is defined [1 ] as follows. With every v on a is associated, by L, a unique single-valued harmonic function Lv on G which satisfies the following conditions: