  We discuss a class of time-dependent Hamilton-Jacobi equations, where an
unknown function of time is intended to keep the maximum of the solution to the
constant value 0. Our main result is that the full problem has a unique
viscosity solution, which is in fact classical. The motivation is a
selection-mutation model which, in the limit of small diffusion, exhibits
concentration on the zero level set of the solution of the Hamilton-Jacobi
equation. Uniqueness is obtained by noticing that, as a consequence of the
dynamic programming principle, the solution of the Hamilton-Jacobi equation is
classical. It is then possible to write an ODE for the maximum of the solution,
and treat the full problem as a nonstandard Cauchy problem.
