A short time existence/uniqueness result for a nonlocal topology-preserving segmentation model

Nicolas Forcadel (1) , Carole Le Guyader (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie

Motivated by a prior applied work of Vese and the second author dedicated to segmentation under topological constraints, we derive a slightly modified model phrased as a functional minimization problem, and propose to study it from a theoretical viewpoint. The mathematical model leads to a second order nonlinear PDE with a singularity at $\nabla u=0$ and containing a nonlocal term. A suitable setting is thus the one of the viscosity solution theory and, in this framework, we establish a short time existence/uniqueness result as well as a Lipschitz regularity result for the solution.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Carole Le Guyader. A short time existence/uniqueness result for a nonlocal topology-preserving segmentation model. Journal of Differential Equations, 2012, 253, pp. 977-995. ⟨hal-00543911⟩