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Archive for Rational Mechanics and Analysis

, Volume 218, Issue 3, pp 1263–1329 | Cite as

Nonlocal Curvature Flows

  • Antonin Chambolle
  • Massimiliano Morini
  • Marcello PonsiglioneEmail author
Article

Abstract

This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. We introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper also allows us to establish new results.

Keywords

Viscosity Solution Comparison Principle Viscosity Subsolution Compact Boundary Geometric Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Antonin Chambolle
    • 1
  • Massimiliano Morini
    • 2
  • Marcello Ponsiglione
    • 3
    Email author
  1. 1.CMAP, Ecole Polytechnique, CNRSPalaiseauFrance
  2. 2.Dip. di MatematicaUniv. ParmaParmaItaly
  3. 3.Dip. di MatematicaUniv. Roma-I “La Sapienza”RomeItaly

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