Γ-convergence for nonlocal phase transitions

Savin, Ovidiu; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 479-500.

We discuss the Γ-convergence, under the appropriate scaling, of the energy functional

$$\parallel u{\parallel }_{H^s\left(\Omega \right)}^2+\underset{\Omega }{\int }W\left(u\right)dx,$$

with $s\in (0,1)$, where $\parallel u{\parallel }_{H^s\left(\Omega \right)}$ denotes the total contribution from Ω in the $H^s$ norm of u, and W is a double-well potential.When $s\in [1/2,1)$, we show that the energy Γ-converges to the classical minimal surface functional – while, when $s\in (0,1/2)$, it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.

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