We study the asymptotic behavior as epsilon goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,E-epsilon(s)(u) = epsilon(2s) integral integral(IxI) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(1+2s) dxdy + integral(I) W(u) dx,where I is an interval in R, and W is a double-well potential. We provide a Gamma-convergence result for any s is an element of (0, 1), by extending the case when s = 1/2 studied by Alberti, Bouchitte and Seppecher in [2]. We also investigate the convergence as s NE arrow 1 of the related optimal profile problem to the local counterpart.

Gamma-Convergence for one-dimensional nonlocal phase transition energies / Palatucci, Giampiero; Vincini, Simone. - In: LE MATEMATICHE. - ISSN 0373-3505. - 75:1(2020), pp. 195-220. [10.4418/2020.75.1.10]

Gamma-Convergence for one-dimensional nonlocal phase transition energies

Palatucci, Giampiero;Vincini, Simone
2020

Abstract

We study the asymptotic behavior as epsilon goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,E-epsilon(s)(u) = epsilon(2s) integral integral(IxI) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(1+2s) dxdy + integral(I) W(u) dx,where I is an interval in R, and W is a double-well potential. We provide a Gamma-convergence result for any s is an element of (0, 1), by extending the case when s = 1/2 studied by Alberti, Bouchitte and Seppecher in [2]. We also investigate the convergence as s NE arrow 1 of the related optimal profile problem to the local counterpart.
Gamma-Convergence for one-dimensional nonlocal phase transition energies / Palatucci, Giampiero; Vincini, Simone. - In: LE MATEMATICHE. - ISSN 0373-3505. - 75:1(2020), pp. 195-220. [10.4418/2020.75.1.10]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2884337
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