We discuss the Γ-convergence, under the appropriate scaling, of the energy functional∥u∥ Hs(Ω) 2+∫ ΩW(u)dx, with s∈(0,1), where ∥u∥H s (Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. When s∈[1/2,1), we show that the energy Γ-converges to the classical minimal surface functional - while, when s∈(0,1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.
Γ-convergence for nonlocal phase transitions / O. Savin, E. Valdinoci. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - 29:4(2012), pp. 479-500.
Γ-convergence for nonlocal phase transitions
E. Valdinoci
2012
Abstract
We discuss the Γ-convergence, under the appropriate scaling, of the energy functional∥u∥ Hs(Ω) 2+∫ ΩW(u)dx, with s∈(0,1), where ∥u∥H s (Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. When s∈[1/2,1), we show that the energy Γ-converges to the classical minimal surface functional - while, when s∈(0,1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.
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