Very Singular Diffusion Equations-Second and Fourth Order Problems

This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example

A variational view at the time-dependent minimal surface equation

We present a global variational approach to the L2-gradient flow of the area functional of cartesian surfaces through the study of the so-called weighted energy-dissipation (WED) functional. In particular, we prove a relaxation result which allows us to show that minimizers of the WED converge in a quantitatively prescribed way to gradient-flow trajectories of the relaxed area functional. The result is then extended to general parabolic quasilinear equations arising as gradient flows of convex functionals with linear growth