The divergence theorem and the Laplacian in Minkowski space
Given a Minkowski space (a finite-dimensional Banach space), the authors define a Minkowskian notion of divergence for 1-forms and prove a corresponding divergence theorem. Since the Euclidean divergence theorem involves unit normal vectors and area integration, for the Minkowski counterpart one needs not only Minkowskian length and a notion of normality (here Birkhoff's definition is used), but also a Minkowskian notion of surface area. This can here be any Minkowskian area satisfying the usual axioms and specified by a convex isoperimetrix. After establishing the divergence for 1-forms, the authors define a Minkowskian Laplace operator by applying the divergence to the differential of a function. It is proved that this Laplacian is a second-order, constant-coefficient, elliptic differential operator. Its symbol is computed and used to assign a natural Euclidean structure with the Minkowski space.
scientific article; zbMATH DE number 942746