  We consider problems of one dimensional interacting fermions confined to a
finite size, multichannel geometry. Concentrating on Luttinger liquids and
carbon nanotubes, we use nontrivial boundary conditions to represent the effect
of external leads, and apply our framework to transport problems in a Josephson
junction setup. We present an exact computation of all correlation functions,
including finite-size and temperature effects, for two sets of solvable
boundary conditions. In all cases, we compute physical quantities like the
Josephson current and the pairing order parameter profile.
