On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

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.

Horizontally quasiconvex envelope in the Heisenberg group

“…We refer to [10,18,22,35,6,20] concerning well-posedness results for this problem. See other related discussions on this topic in [11,13,17].…”