  We investigate a general elliptic problem given in a bounded Euclidean domain
with boundary data in Nikolskii spaces of low, specifically, negative order.
The right-hand side of the elliptic differential equation is supposed to be an
integrable function. We establish the Fredholm property of the problem, the
maximal regularity and a priori estimate of its generalized solutions in the
spaces indicated. We give an application of these results to some elliptic
problems with boundary conditions induced by a Gaussian white noise.
