In questa Nota presentiamo alcuni teoremi di confronto tra il movimento secondo la curvatura media ottenuto con il metodo delle minime barriere di De Giorgi e i movimenti definiti con i metodi di Evans-Spruck, Chen-Giga-Goto, Giga-Goto-Ishii-Sato.
The a priori estimate of the maximum modulus of the generalized solution is established for a doubly nonlinear parabolic equation with special structural conditions.
We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.
The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation , where , and is a convex function of with linear growth as , satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.
We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the...
In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind the Harnack...
Modelling of macroscopic behaviour of materials, consisting of several layers or components, cannot avoid their microstructural properties. This article demonstrates how the method of Rothe, described in the book of K. Rektorys The Method of Discretization in Time, together with the two-scale homogenization technique can be applied to the existence and convergence analysis of some strongly nonlinear time-dependent problems of this type.