A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients

Zan, Cecilia De; Soravia, Pierpaolo

doi:10.1155/2016/3627896

Cited by 2 publications

(1 citation statement)

References 20 publications

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“…However, the standard proof does not seem to apply with f e 6 ¼ 0 and we need a continuity of u or v to recover (3.4). The proof of Proposition 3.2 uses the idea in [11]. We need (3.4) to show that the left-hand side of (3.8) below converges to zero as e !…”

Section: Comparison Principlementioning

confidence: 99%

Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

Giga

Požár

2020

SN Partial Differ. Equ. Appl.

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A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

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Section: Comparison Principlementioning

confidence: 99%

Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

Giga

Požár

2020

SN Partial Differ. Equ. Appl.

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On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Zan

Soravia

2020

Abstract and Applied Analysis

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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.

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A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients

Cited by 2 publications

References 20 publications

Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

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