A caricature of a singular curvature flow in the plane

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in details the facet formation and interactions and their asymptotic behavior. We notice that our qualitative results may be interpreted with the help of a special composition of multivalued operators

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

Berg's effect

A Neumann problem for the Laplace equation is considered outside a three dimensional straight cylinder. The value of a solution O" at space infinity is prescribed. The Neumann data aO" / an ( n is the outer normal of the cylinder) is assumed to be independent of the spatial variables on the top and the bottom and also on the lateral part of the boundary of the cylinder. The behavior of the value of O" on the boundary is studied. In particular, it is shown that O" is an increasing function of the distance from the center of the top ( respectively, the bottom) if a(J" / an > o on the lateral part and a(J" / an is the same constant on the top and (respectively, the bottom). An analogous statement is shown for O" on the lateral part. In the theory of crystal growth O" is interpreted as a supersaturation and cylinder is a crystal. The value aO" / an is the growth speed. The main contribution of this paper is considered as the first rigorous proof of Berg's effect when the crystal shape is a cylinder

Energy solutions to one-dimensional singular parabolic problems with BVBV data are viscosity solutions

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.Comment: 15 page

Helicoidal surfaces with constant anisotropic mean curvature

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature surfaces can be obtained by quadratures

On instant extinction for very fast diffusion equations

In this paper we prove instant extinction of the solutions to Dirichlet and Neumann boundary value problem for some quasilinear parabolic equations whose diffusion coefficient is singular when the spatial gradient of unknown function is zero