A Dirichlet mapping
between regions in Euclidean space is a homeomorphism preserving the finiteness of
Dirichlet integrals of admissible functions and plays an important role in the
potential theory. Two dimensional Dirichlet mappings are known to be characterized
geometrically as being quasiconformal mappings. In this paper, higher dimensional
Dirichlet mappings will be characterized geometrically as being quasi-isometries. In
order to carry out the reasoning it is necessary to study the relation between the
Radon-Nikodym density R and the Jacobian J of an arbitrary homeomorphism for
which only existences of R and J almost everywhere are assured. It will be
proven that R ≦|J|, almost everywhere, which is the main result of this
paper.
