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.

Nonlocal curvature flows / Chambolle, Antonin; Morini, Massimiliano; Ponsiglione, Marcello. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - 218:3(2015), pp. 1263-1329. [10.1007/s00205-015-0880-z]

Nonlocal curvature flows

MORINI, MASSIMILIANO;PONSIGLIONE, Marcello
2015

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.
2015
analysis; mathematics (miscellaneous); mechanical engineering
01 Pubblicazione su rivista::01a Articolo in rivista
Nonlocal curvature flows / Chambolle, Antonin; Morini, Massimiliano; Ponsiglione, Marcello. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - 218:3(2015), pp. 1263-1329. [10.1007/s00205-015-0880-z]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/979568
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