The Kobayashi–Warren–Carter energy is diffusive interfacial energy, which considers the structures of an average orientation of each phase. In a one-dimensional setting, its sharp interface limit in the sense of the Gamma-convergence is given with a heuristic explanation. One key point is that one must use a notion of convergence finer than conventional convergence for order parameters. The limit sharp interfacial energy is a kind of total variation; however, it has different properties. The existence of a minimizer of the limit interfacial energy is established. If one minimizes this energy in the order parameter, a kind of total variation energy is obtained. This modified total variation energy avoids a small staircase structure, which causes a severe problem when the total variation energy is used for image denoising. Furthermore, this paper studied the stationary points of the modified total variation energy for piecewise monotone function with only one jump.

In this paper, we establish the existence and uniqueness of a solution to a double barrier backward doubly stochastic differential equation (DB-BDSDE). This would be proved by using the “penalization method” as in the case of a single barrier backward doubly stochastic differential equation. Additionally, we give a connection between DB-BDSDE and obstacle stochastic partial differential equation (SPDE). To achieve this, we claim that through the existence of a stochastic viscosity solution to obstacle SPDE associated with DB-BDSDE using the so-called Doss–Sussmann transformation.