Nonlocal Curvature Flows

By Antonin Chambolle, Massimiliano Morini, Marcello Ponsiglione. Archive for Rational Mechanics and Analysis, 2014

102 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79644c3d671de58d047b8609c2f30f35e0fab331

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Hölder regularity for nonlocal double phase equations (2019, methodology): “For this, some very important tools recently introduced in the nonlocal theory, as the celebrated Caffarelli-Silvestre s-harmonic extension [8], and other minor successful tricks, as for instance the pseudo-differential commutator compactness in [31] or the energy density estimates in [10,19,32], se”