Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. S. Gelli - S. Luckhaus - M. Novaga

Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model

created by novaga on 21 Jul 2005
modified on 04 Apr 2016


Published Paper

Inserted: 21 jul 2005
Last Updated: 4 apr 2016

Journal: Calc. Var. Partial Differential Equations
Volume: 26
Number: 4
Pages: 429-445
Year: 2006


For the Allen-Cahn functional we study the following problem: for which prescribed amount $m$ of volume is there the appearence of a droplet of one phase inside the other? Under a suitable assumption on the domain we show that the breaking of symmetry occurs at the same value of $m$ as for the limit of the sharp interface energy. We also prove that there exists a threshold for $m$ of order $\epsilon^{n/(n+1)}$ so that either there is the appearence of the droplet or there is no breaking of symmetry.