On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Cecilia De Zan,Pierpaolo Soravia

doi:10.1155/2020/7545983

Abstract

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

Highlights

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. ese are characteristic points for the level sets of the solutions and are usually difficult to deal with
We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present
As a matter of fact, we will use this notion of solution in a forthcoming paper, when we discuss the singular limit of reaction diffusion equations for anisotropic and degenerate diffusions [6], while we develop here the preliminary needed tools on weak front propagation. is simplified approach, which is helpful when studying approximations of (1) of different nature, extends to the Carnot group setting with similar properties

Summary

Introduction

We want to discuss the notion of viscosity solution for geometric equations, describing weak front propagation in step two Carnot groups, of the form ut(x, t) + F􏼐x, t, Xu(x, t), X2u(x, t)􏼑 0, (1). Is equivalent notion of solution simplifies the dealing with singularities and was first proved in the Euclidian setting for the mean curvature flow equation by Barles and Georgelin [3] to study the convergence of numerical schemes. If in particular we consider the recent notion of v-convex functions with respect to the family of vector fields, we can prove, coupling our result with a comparison principle, that their level sets become extinct in finite time under the horizontal mean curvature flow equation, by constructing suitable supersolutions of (1). One of the referees pointed out to us the work of Ferrari et al [12] where the authors use an approach similar to ours in the case of the horizontal mean curvature flow equation in the Heisenberg group, and they show a comparison principle for axisymmetric viscosity solutions. For the mathematical analysis of the level set method via viscosity solutions, the reader is referred to the book by Giga [15], where the approach is discussed in detail (see Souganidis [16] and the references therein for the main applications of the theory, and [17] for equations with discontinuous coefficients)

Step Two Carnot Groups and Level Set Equations on the Group

Viscosity Solutions

Examples of Explicit Supersolutions or Subsolutions

A Geometric Definition of Generalised Flow in Carnot Groups