  We study the local regularity of sliding almost minimal sets of dimension 2
in $R^n$ , bounded by a smooth curve $L$. These are a good way to model soap
films bounded by a curve, and their definition is similar to Almgren's. We aim
for a local description, in particular near L and modulo $C^{1+$\epsilon$}$
diffeomorphisms, of such sets $E$, but in the present paper we only obtain a
full description when $E$ is close enough to a half plane, a plane or a union
of two half planes bounded by the same line, or a transverse minimal cone of
type $Y$ or $T$. The main tools are adapted near monotonicity formulae for the
density, including for balls that are not centered on L, and the same sort of
construction of competitors as for the generalization of J. Taylor's regularity
result far from the boundary.
