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.
Bellettini, G., Gelli, M.S., Luckhaus, S., Novaga, M. (2006). Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 26(4), 429-445 [10.1007/s00526-006-0012-6].
Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model
BELLETTINI, GIOVANNI;
2006-01-01
Abstract
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.
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https://hdl.handle.net/11365/1017419