  Recent years witnessed an extensive development of the theory of the critical
point in two-dimensional statistical systems, which allowed to prove {\it
existence} and {\it conformal invariance} of the {\it scaling limit} for
two-dimensional Ising model and dimers in planar graphs. Unfortunately, we are
still far from a full understanding of the subject: so far, exact solutions at
the lattice level, in particular determinant structure and exact discrete
holomorphicity, play a cucial role in the rigorous control of the scaling
limit. The few results about not-integrable (interacting) systems at
criticality are still unable to deal with {\it finite domains} and {\it
boundary corrections}, which are of course crucial for getting informations
about conformal covariance. In this thesis, we address the question of adapting
constructive Renormalization Group methods to non-integrable critical systems
in $d= 1+1$ dimensions. We study a system of interacting spinless fermions on a
one-dimensional semi-infinite lattice, which can be considered as a prototype
of the Luttinger universality class with Dirichlet Boundary Conditions. We
develop a convergent renormalized expression for the thermodynamic observables
in the presence of a quadratic {\it boundary defect} counterterm, polynomially
localized at the boundary. In particular, we get explicit bounds on the
boundary corrections to the specific ground state energy.
