On the Complete Representation of Biharmonic Functions
Roger L. Fosdick · SIAM Journal on Applied Mathematics · 1970
The emphasis of this paper is on the complete representation of biharmonic functions in terms of harmonic functions for arbitrary bounded three-dimensional domains. The main result is contained in Theorem 2, which allows for the possibility that the domain may possess inclusions (i.e., holes). Reduced forms of the representation are noted under restrictive hypotheses on the geometry of the domain. A brief analogous treatment for plane two-dimensional domains is considered in order to illustrate the significant difference that here similar reduced forms are complete without geometric restrictions on the region. However, in this case, multivalued harmonic functions are generally necessary.