| Abstrakt: |
In this chapter, we show how one can apply the variational method to the study of the theory of harmonic integrals. In his first paper on the subject, W. V. D. Hodge [1] used a variational method in this theory to study certain boundary value problems for forms defined on domains in Euclidean space (using Cartesian coordinates). But, in order to carry his theory over to compact Riemannian manifolds, he and subsequent writers found it expedient to employ methods involving integral equations (see Hodge [2], Kodaira, De Rham-Kodaira, and references therein). More recently, Milgram and Rosenbloom ([1], [2]) and Gaffney ([1], [2]), have treated certain problems by their ˵heat equation″ method involving parabolic equations. The variational method was applied to general compact Riemannian manifolds by Morrey and Eells and to such manifolds with boundary by Morrey [11]. A closely related method was employed concurrently by Friedrichs [3] in both cases. Certain boundary value problems had been discussed previously by Duff and Spencer and by Connor ([1], [2]). [ABSTRACT FROM AUTHOR] |
