Biharmonic functions with prescribed fine normal derivative on the Martin boundary (Q1100588)

The Dirichlet problem for the bi-Laplacian considered here is that of finding a function u on an open subset R of n-dimensional Euclidean space having continuous fourth partial derivatives such that (i) \(\Delta^ 2u=0\) on R, where \(\Delta\) is the Laplacian, (ii) \(-lim_{\xi \to \eta}u(\xi)=0\) a.e. on the Martin boundary \(R_ M\) of R, where f-lim indicates the limits of functions on \(R\cup R_ M\) in the fine topology, and (iii) the ``fine normal derivative,'' i.e. \(-lim_{\xi \to \eta}(u(\xi)- u(\eta))/G(\xi_ 0,\xi)\) equals f a.e. on \(R_ M\), where f is some element in a rather general class of functions on \(R_ M\), and where G is the Green function on \(R\times R.\) The author proves two theorems concerning the existence of solutions to this Dirichlet problem.