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2020 On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations
Cecilia De Zan, Pierpaolo Soravia
Abstr. Appl. Anal. 2020: 1-14 (2020). 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.

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Cecilia De Zan. Pierpaolo Soravia. "On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations." Abstr. Appl. Anal. 2020 1 - 14, 2020. https://doi.org/10.1155/2020/7545983

Information

Received: 30 July 2019; Revised: 1 October 2019; Accepted: 28 October 2019; Published: 2020
First available in Project Euclid: 27 May 2020

zbMATH: 07176264
MathSciNet: MR4053149
Digital Object Identifier: 10.1155/2020/7545983

Rights: Copyright © 2020 Hindawi

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