STABLE AND UNSTABLE MANIFOLDON A FREE BOUNDARY PROBLEM OFTHE CURVATURE FLOW WITH DRIVING FORCE

We study a free boundary problem for the curvature ow with driving force for a family of planar curves with two xed contact angles on the x-axis. Our aim of this paper is to analyze the dimension of stable and unstable manifolds of traveling wave solution

Mid-infrared spectroscopic investigation of the perfect vitrification of poly(ethylene glycol) aqueous solutions.

Crystallization/recrystallization behaviors of poly(ethylene glycol) (PEG) aqueous solutions with water contents (WC\u27s) of ∼36-51 wt % were investigated by temperature-variable mid-infrared spectroscopy. At a WC of 43.2 wt %, crystallization and recrystallization of water and PEG were not observed. At this specific WC value (WCPV), perfect vitrification occurred. Below and above the WCPV value, crystallization/recrystallization behaviors changed drastically. The crystallization temperature below WCPV (237 K) was ∼10 K greater than that above WCPV (226 K). Recrystallization above and below WCPV occurred in one (213 K) and two (198 and 210 K) steps, respectively. These findings resulted from the difference in the (re)crystallization behaviors of water molecules associated with PEG chains with helical and random-coil conformations. These two types of water molecules might have limiting concentrations for their (re)crystallization, indicating that perfect vitrification might have occurred when the concentrations of the two types of water molecules were less than the limiting concentrations of their (re)crystallization

Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures

Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces

Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface

A method is presented for generating a discrete piecewise constant Gaussian curvature (CGC) surface. An energy functional is first formulated so that its stationary point is the linear Weingarten (LW) surface, which has a property such that the weighted sum of mean and Gaussian curvatures is constant. The CGC surface is obtained using the gradient derived from the first variation of a special type of the energy functional of the LW surface and updating the surface shape based on the Gaussian curvature flow. A filtering method is incorporated to prevent oscillation and divergence due to unstable property of the discretized Gaussian curvature flow. Two techniques are proposed to generate a discrete piecewise CGC surface with preassigned internal boundaries. The step length of Gaussian curvature flow is adjusted by introducing a line search algorithm to minimize the energy functional. The effectiveness of the proposed method is demonstrated through numerical examples of generating various shapes of CGC surfaces

Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations

This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result