  The formation of codimension-one interfaces for multi-well gradient-driven
problems is well-known and established in the scalar case, where the equation
is often referred to as the Allen-Cahn equation. The proofs rely for a large on
a monotonicity formula for the energy density, which is itself related to the
vanishing of the so-called discrepancy function. The vectorial case in contrast
is quite open. This lack of results and insight is to a large extend related to
the absence of known appropriate monotonicity formula. In this paper, we focus
on the \emph{elliptic case in two dimensions}, and introduce methods, relying
on the analysis of the partial differential equation, which allow to circumvent
the lack of monotonicity formula for the energy density. In the last part of
the paper, we recover a \emph{new monotonicity formula} which relies on a
\emph{new discrepancy relation}. These tools allow to extend to the vectorial
case in two dimensions most of the results obtained for the scalar case. We
emphasize also some \emph{specific features} of the vectorial case.
