Integrality of varifolds in the singular limit of reaction-diffusion equations

Yoshihiro Tonegawa
2003 Hiroshima Mathematical Journal  
We answer a question posed by Ilmanen on the integrality of varifolds which appear as the singular perturbation limit of the Allen-Cahn equation. We show that the density of the limit measure is integer multiple of the surface constant almost everywhere at almost all time. This shows that limit measures obtained via the Allen-Chan equation and those via Brakke's construction share the same integrality property as well as being weak solutions for the mean curvature flow equation. 2000
more » ... on. 2000 Mathematics Subject Classification. 35K57, 35B25, 49Q20.
doi:10.32917/hmj/1150997978 fatcat:bmegabs5u5hm7m3w6emc3osceq