For certain elliptic differential operators L, we study the behaviour of solutions to \(Lu=0,\) as we tend to the boundary along radii in strictly starlike domains in \(\mathbb {R}^n, n\ge 3.\) Analogous results are obtained in other special domains. Our approach involves introducing harmonic line bundles as instances of Brelot harmonic spaces and approximating continuous functions by harmonic functions on appropriate subsets. We are required to approximate on certain closed sets, which is not obvious, since the space of continuous functions on an (unbounded) closed set, endowed with the topology of uniform convergence, is not a topological vector space, though it is both a vector space and a topological space.
