  In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations
appear naturally in the asymptotic limit of selection-mutation models when the
population variance vanishes. They have to be solved together with an unknown
function I(t) that arises as the counterpart of a non-negativity constraint on
the solution at each time. Although the uniqueness of viscosity solutions is
known for many variants of Hamilton-Jacobi equations, the uniqueness for this
particular type of constrained problem was not resolved, except in a few
particular cases. Here, we provide a general answer to the uniqueness problem,
based on three main assumptions: convexity of the Hamiltonian function H(I, x,
p) with respect to p, monotonicity of H with respect to I, and BV regularity of
I(t).
