The total variation flow is by now very popular for image denoising and also in materials science. When we model relaxation of crystal surfaces below the roughening temperature, the fourth-order total variation flow is often used. However, compared with the second-order problem, its analytic property has not been well studied because the definition of the solution itself is not trivial.

In this talk, we define rigorously its solution in the whole Euclidean space R^n. It turns out that if n is greater than 2 it can be understood as a gradient flow of the total variation energy in D^{-1},the dual space of D^1_0, which is a completion of the space of compactly supported smooth function in the Dirichlet norm. However, if n is less than or equal to 2, the space D^{-1} imposed average free condition at least formally. Since we are interested in motion of characteristic functions, we extend the notion of a solution in a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a kind of duality argument.

We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout evolution.

We prove that all balls are calibrable. However, unlike in the second order problem, the outside a ball is not calibralbe if and only if n=2. If n is not 2, all annuli are calibrable, while in the case n=2, it is not calibrable if an annulus is too thick. We give a whole evolution of the characteristic function of a ball. We even study evolution of several piecewise constant radial functions. This is a joint work of Hirotoshi Kuroda (Hokkaido University) and Michał Łasica (Polish Academy of Sciences).