We study the motion of regular bent rectangles driven by singular curvature flow with a driving term. The curvature is being interpreted as a solution to a minimization problem. The evolution equation becomes in a local coordinate a system of Hamilton--Jacobi equations with free boundaries, coupled to a system of ODE's with nonlocal nonlinearities. We establish local-in-time existence of variational solutions to the flow, and uniqueness is proved under additional regularity assumptions on the data.

In this paper we study blow-up phenomena of the solutions of a reaction-diffusion equation for which the diffusion term corresponds to the diffusion equation in transparent media. We show that for this equation the critical Fujita exponent is smaller than the classical one.

In this paper we investigate the motion of one-dimensional graphs under anisotropic non-convex mean curvature flow regularized via a Willmore term. Aiming at understanding the evolution problem when we let the regularization parameter tend to zero, we first present rigorous analytical results for the stationary case. Subsequently we discuss the time-dependent problem, focussing mainly on numerical simulations. We discretize by finite elements, and provide a semi-implicit scheme and a number of numerical experiments.

In this paper, we consider optimal control problems for one-dimensional phase field system with total variation functional as the interfacial energy. Our system consists of two parabolic PDEs: a heat equation and a singular diffusion equation of an order parameter. We prove the existence of an optimal control that minimizes the nonlinear and nonsmooth cost functional. Moreover, we show the necessary condition of the optimal pair by using the optimal control problem of the approximating system.

In this paper, variational inclusions of Euler--Lagrange types, governed by two-dimensional isotropic-anisotropic singular diffusions, are considered. On that basis, we focus on the geometric structures of free boundaries where anisotropic conditions tend to isotropic. In this light, a limit-set of special piecewise-constant solutions will be presented. The objective in this paper is to give some observations on the upper bounds of the limit set with geometric characterizations. As a consequence, it will be shown that the isotropic free boundaries, as in the limit set, consist of a finite number of $ C^{1,1} $-Jordan curves, and these have certain geometric connections with the approaching anisotropic situations. Observations for the lower bounds will be studied in the sequel to this paper.

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite-element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties, which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.