Symmetric harmonic sheaves possessing bipotentials (Q1398032)

Let \(\mathcal{H}\) be a family of sheaves \(H\) of continuous functions on a locally compact space with a countable base such that locally the Dirichlet problem with respect to \(H\in\mathcal{H}\) is solvable, \(H\) satisfies Harnack inequalities and has some additional symmetry property. The authors define the notions of \(H\)-biharmonic functions, \(H\)-biharmonic Green functions, \(H\)-biharmonic potentials, etc. and study the interrelation between them. This consideration leads to a classification of the family \(\mathcal{H}.\) The authors give some particular applications of their results. For example they get a new necessary and sufficient condition for the existence of a classical biharmonic Green function for the Laplace-Beltrami operator \(\Delta\) on a Riemannian manifold. Further applications are also given.