  This work addresses the question of uniqueness of the minimizers of a convex
but not strictly convex integral functional with linear growth in a
two-dimensional setting. The integrand -- whose precise form derives directly
from the theory of perfect plasticity -- behaves quadratically close to the
origin and grows linearly once a specific threshold is reached. Thus, in
contrast with the only existing literature on uniqueness for functionals with
linear growth, that is that which pertains to the generalized least gradient,
the integrand is not a norm. We make use of hyperbolic conservation laws hidden
in the structure of the problem to tackle uniqueness. Our argument strongly
relies on the regularity of a vector field -- the Cauchy stress in the
terminology of perfect plasticity -- which allows us to define characteristic
lines, and then to employ the method of characteristics. Using the detailed
structure of the characteristic landscape evidenced in our preliminary study
\cite{BF}, we show that this vector field is actually continuous, save for
possibly two points. The different behaviors of the energy density at zero and
at infinity imply an inequality constraint on the Cauchy stress. Under a
barrier type convexity assumption on the set where the inequality constraint is
saturated, we show that uniqueness holds for pure Dirichlet boundary data, a
stronger result than that of uniqueness for a given trace on the whole boundary
since our minimizers can fail to attain the boundary data.
