  Consider the viscous Burgers equation on a bounded interval with
inhomogeneous Dirichlet boundary conditions. Following the variational
framework introduced by Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim C, we
analyze a Lyapunov functional for such equation which gives the large
deviations asymptotics of a stochastic interacting particles model associated
to the Burgers equation. We discuss the asymptotic behavior of this energy
functional, whose minimizer is given by the unique stationary solution, as the
length of the interval diverges. We focus on boundary data corresponding to a
standing wave solution to the Burgers equation in the whole line. In this case,
the limiting functional has in fact a one-parameter family of minimizers and we
analyze the so-called development by Gamma-convergence; this amounts to compute
the sharp asymptotic cost corresponding to a given shift of the stationary
solution.
