DISCONTINUOUS STATIONARY SOLUTION TO GENERALIZED EIKONAL-CURVATURE EQUATION AND ITS STABILITY

In this paper we consider stability of stationary solutions to an eikonal-curvature ow equation (1.1) V = C +K on t for evolving interface t in a domai

On a uniform approximation of motion by anisotropic curvature by the Allen-Cahn equations

The convergence of solutions of anisotropic Allen-Cahn equations is studied when the interface thickness parameter(denoted by $\varepsilon$) tends to zero. It is shown that the convergence to a level set solution of the corresponding anisotropic interface equations is uniform with respect to the derivatives of a suface energy density function. As an application a cryatalline motion of interfaces in shown to be approximated by anisotropic Allen-Cahn equations

A LEVEL SET APPROACH REFLECTING SHEET STRUCTURE WITH SINGLE AUXILIARY FUNCTION FOR EVOLVING SPIRALS ON CRYSTAL SURFACES

We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal-curvature ow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary function u minus a pre-determined multi-valued sheet structure function , which re ects the strength of spirals (screw dislocation centers). The level set equation used in our method for u is the same as that of the eikonal-curvature ow equation. The multi-valued nature of the sheet structure function is only invoked when preparing the initial auxiliary function, which is nontrivial, and in the nal step when extracting information such as the height of the spiral steps. Our simulation enables us not only to reproduce all speculations on spirals in a classical paper by Burton, Cabrera and Frank (1951) but also to nd several new phenomena

On behavior of signs for the heat equation and a diffusion method for data separation

Consider the solution u(x; t) of the heat equation with initial data u0. The diffusive sign SD[u0](x) is de ned by the limit of sign of u(x; t) as t ! 0. A sufficient condition for x 2 Rd and u0 such that SD[u0](x) is well-de ned is given. A few examples of u0 violating and ful lling this condition are given. It turns out that this diffusive sign is also related to variational problem whose energy is the Dirichlet energy with a delty term. If initial data is a difference of characteristic function of two disjoint sets, it turns out that the boundary of the set SD[u0](x) = 1 (or 1) is roughly an equi-distance hypersurface from A and B and this gives a separation of two data sets

OPTIMAL CONTROL OF A SINGULAR DIFFUSION EQUATION WITH CONSTRAINT

In this paper, we consider an optimal control problem for a state problem of a one dimensional Allen-Cahn type equation, which is derived from a free energy including the total variation interfacial energy and the constraint by the indicator function. So, the state problem contains the singular diffusion coming from the total variation, and another singular term coming from the constraint. The main conclusions of this paper are concerned with the existence of an optimal control (optimal pair) and the necessary condition (optimality condition) of an optimal pair. Furthermore, the analytic methods, performed in this paper, will provide an effective approximating approach to the optimal pairs of our control problem

Hepatocyte apoptosis is enhanced after ischemia/reperfusion in the steatotic liver

Liver steatosis is associated with organ dysfunction after hepatic resection and transplantation which may be caused by hepatic ischemia/reperfusion injury. The aim of the current study was to determine the precise mechanism leading to hepatocyte apoptosis after steatotic liver ischemia/reperfusion. Using a murine model of partial hepatic ischemia for 90 min, we examined the levels and pathway of apoptosis, and the peroxynitrite expression, serum alanine aminotransferase levels, and liver histology 1 and 4 h after reperfusion. In the steatotic liver, the peroxynitrite expression increased after ischemia/reperfusion. Significant hepatocyte apoptosis in the steatotic liver was seen after reperfusion, caused by upregulation of cleaved caspases 9 and 3, but not caspase 8. Serum alanine aminotransferase levels were elevated and histological examination revealed severe liver injury in the steatotic liver 4 h after reperfusion. In mice treated with aminoguanidine, ischemia/reperfusion-induced increases in serum alanine aminotransferase levels and apoptosis were significantly reduced in steatotic liver compared with mice treated with phosphate buffered saline. Survival of mice with steatotic livers significantly improved by treatment with aminoguanidine. Our data suggested that the steatotic liver is vulnerable to hepatic ischemia/reperfusion, leading to significant hepatocyte apoptosis by the mitochondrial permeability transition, and thereby resulting in organ dysfunction