  We discuss the two-dimensional Grassmannian $SU(N)/S(U(N-2)\times U(2))$ and
the flag $SU(N)/S(U(N-2)\times U(1)\times U(1))$ sigma models on a finite
interval and construct analytical solutions of gap equations in the large N
limit. We show that the flag model admits a homogeneous solution for `mixed'
Dirichlet-Neumann (DN) boundary conditions only for sufficiently large length
$L$ and undergoes a phase transition from the phase of partly broken gauge
symmetry ($U(1)$) to the symmetric phase ($U(1)\times U(1)$) for large $L$. On
the other hand, the Grassmannian model has a detached phase with one massive
and one massless non-zero condensates that completely break $U(2)$ gauge
symmetry. This phase lives on a region of $L$ bounded from above and has to use
the Robin boundary conditions. We also examine the L-dependence of the total
energy and detect the linear growth inherent to confining string in all phases.
