On viscosity and equivalent notions of solutions for anisotropic geometric equations

TL;DR

This paper establishes an equivalence for viscosity solutions of geometric equations in Carnot groups by restricting test functions at singular points, simplifying analysis of singularities and extending generalized flow concepts.

Contribution

It introduces a new approach to handle singular points in Carnot groups, extending Euclidean methods to these non-Euclidean structures.

Findings

Simplifies the analysis of singularities in geometric equations.

Shows boundaries of convex sets become extinct under horizontal mean curvature flow.

Extends the definition of generalized flow to Carnot groups.

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 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.