Robinsches und pragersches problem der potentialtheorie bei beliebiger dimension

Abstract

The Robin problem of potential theory in three dimensions is concerned with the representation of circulation-free vector fields, given in a region B⊂R³, by distributions of sources on the region B and its boundary [d]B in the sense of the Newton-Coulomb law; the Parger problem is concerned with the representation of source-free vector fields by distributions of vortices in the sense of the Biot-Savart law. The Robin problem a corresponding result on the Prager problem. The aim of this paper is to generalize the Robin and Prager representation problems to skew-symmetric tensor fields of arbitrary rank in the Euclidean space of arbitrary dimension, and to obtain a better understanding of the duality of these problems. The main results are constructive proofs of the existence and uniqueness of the solutions of the general Robbin and Prager problems. This is done by reducing these representation problems to equivalent integral equation formulations and applying Fredholm's alternative. For this recent results of the author on the Dirichlet and Neumann problems is governed by the so-called Hodge duality operator: theorems on the Prager problem are obtained from corresponding theorems on the Robin problem by applying the Hodge duality operator.