  Patterns of symmetry breaking induced by potentials at the boundary of free
$O(N)$ models in $d=3- \epsilon$ dimensions are studied. We show that the
spontaneous symmetry breaking in these theories leads to a boundary RG flow
ending with $N - 1$ Neumann modes in the IR. The possibility of
fluctuation-induced symmetry breaking is examined and we derive a general
formula for computing one-loop effective potentials at the boundary. Using the
$\epsilon-$expansion we test these ideas in an $O(N)\oplus O(N)$ model with
boundary interactions. We determine the RG flow diagram of this theory and find
that it has an IR-stable critical point satisfying conformal boundary
conditions. The leading correction to the effective potential is computed and
we argue the existence of a phase boundary separating the region flowing to the
symmetric fixed point from the region flowing to a symmetry-broken phase with a
combination of Neumann and Dirchlet boundary conditions.
