Minimum principles for the trajectories of systems governed by rate problems

Recently, Mielke and Ortiz [2007. A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var., in press] have proposed a variational reformulation of evolutionary problems that characterizes entire trajectories of a system as minimizers of certain energy–dissipation functionals. In this paper we present two examples of energy–dissipation functionals for which relaxations and optimal scalings can be rigorously derived. The first example concerns a one-dimensional bar characterized by a quadratic dissipation function and a bistable energy density; the second example concerns the coarsening kinetics of island growth in thin films exhibiting a preferred slope. In both cases, we present closed-form relaxations in the local limit of the problem and optimal scaling relations for the nonlocal problems. The relaxations rigorously characterize macroscopic properties of complex microstructural evolution by means of well-posed effective problems. The scaling relations rigorously characterize some average properties of the coarsening kinetics of the systems and lead to predictions on the growth exponents.