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 [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates / A. Pisante, F. Punzo. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 15:s. i.(2016), pp. 309-341.

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates

F. Punzo
Ultimo
2016

Abstract

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 [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in Riemannian manifolds.
phase-transitions; level sets; geometrical evolution; generalized motion; gradient theory; singular limit; convergence; interfaces; equation; hypersurfaces
Settore MAT/05 - Analisi Matematica
feb-2014
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/254484
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