  For a heat equation with Robin's boundary conditions which depends on a
parameter $\alpha>0$, we prove that its unique weak solution $\rho^\alpha$
converges, when $\alpha$ goes to zero or to infinity, to the unique weak
solution of the heat equation with Neumann's boundary conditions or the heat
equation with periodic boundary conditions, respectively. To this end, we use
uniform bounds on a Sobolev norm of $\rho^\alpha$ obtained from the
hydrodynamic limit of the symmetric slowed exclusion process, plus a careful
analysis of boundary terms.
