Regularity of soap films near the boundary

We try a first description of soap films \$E\$ (more precisely, almost minimal sets of dimension \$2\$) attached to a smooth curve \$L\$ (for instance, by ways of a "sliding boundary condition" where continuous deformation of \$E\$ that preserve \$L\$), near the boundary \$L\$. The idea is to get a local \$C^1\$ description of \$E\$ near \$L\$, depending on the blow-up limits of \$E\$ at \$x \in L\$.