We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial conditions we show non- positivity of the limiting energy discrepancy. This in turn allows us to prove an almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. These results will be used in  to extend previous important results from  in Euclidean space, showing convergence of solutions to the parabolic Allen- Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.
|Titolo:||Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates|
|Autori interni:||PUNZO, FABIO|
|Data di pubblicazione:||2016|
|Rivista:||ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE|
|Appare nelle tipologie:||01.1 Articolo in Rivista|