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Teoremi di confronto tra diverse nozioni di movimento secondo la curvatura media

Giovanni Bellettini, Maurizio Paolini (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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 area preserving curve shortening flow with Neumann free boundary conditions

Elena Mäder-Baumdicker (2015)

Geometric Flows

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 Cauchy problem for a strongly degenerate quasilinear equation

F. Andreu, Vicent Caselles, J. M. Mazón (2005)

Journal of the European Mathematical Society

We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation u t = div 𝐚 ( u , D u ) , where 𝐚 ( z , ξ ) = ∇ ξ f ( z , ξ ) , and f is a convex function of ξ with linear growth as ∥ ξ ∥ → ∞ , 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.

The CUDA implementation of the method of lines for the curvature dependent flows

Tomáš Oberhuber, Atsushi Suzuki, Vítězslav Žabka (2011)

Kybernetika

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...

The Geometry of Differential Harnack Estimates

Sebastian Helmensdorfer, Peter Topping (2011/2012)

Séminaire de théorie spectrale et géométrie

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...

The method of Rothe and two-scale convergence in nonlinear problems

Jiří Vala (2003)

Applications of Mathematics

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.

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