Viscosity solutions for a polymer crystal growth model

Pierre Cardaliaguet (1) , Olivier Ley (2) , Aurélien Monteillet (1)

1 LM - Laboratoire de mathématiques de Brest
2 IRMAR - Institut de Recherche Mathématique de Rennes

We prove existence of a solution for a polymer crystal growth model describing the movement of a front $(\Gamma(t))$ evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source $\delta_\Gamma$. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with H\"{o}lder bounds in space. From this result, we deduce \textit{a priori} regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation.

Mots clés: Nonlocal Hamilton-Jacobi Equations; heat equation; nonlocal front propagation; level-set approach; geometrical properties; lower-bound gradient estimate; viscosity solutions; eikonal equation; heat equation.

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Cardaliaguet, Olivier Ley, Aurélien Monteillet. Viscosity solutions for a polymer crystal growth model. Indiana University Mathematics Journal, 2011, 60 (3), pp.895-936. ⟨hal-00461361⟩