The Conservation Property of the Heat Equation on Riemannian Manifolds

摘要

The following prohlem has been posed to us by Dr. Edward Nelson:to extend to open Riemannian manifolds (and comoact carried initial functions) the conservation property possessed by solutions of the heat equation on closed (Riemannian) manifolds,nThe heat equation for variable conductivity and heat capacity is discussed in section 2. In section 3 a bounded gradient, c1 approximation to the global distance function is accomplished with the Friedrichs mollifior. In section 4 the conservation property is established for a class of manifolds, permitting infinits volume.