Abstract. In this paper we are interested in the finite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. We show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighbourhood of transition solutions local instabilities occur in finite time. We also show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time.

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. Like solutions of the Cahn-Hilliard equation converging to two main values during the phase separation process, the grayvalues inside the missing part of the image are oriented towards the binary states black and white. We present stability/instability results for solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. In particular we will consider the Willmore functional as a quantity to find the optimal scale of the inpainting