We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional ∥u∥2Hs(Ω)+∫ΩW(u)dx, where ∥u∥Hs(Ω) denotes the total contribution from Ω in the H s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space Rn . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.
Local and global minimizers for a variational energy involving a fractional norm / G. Palatucci, O. Savin, E. Valdinoci. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 192:4(2013), pp. 673-718.
Local and global minimizers for a variational energy involving a fractional norm
E. ValdinociUltimo
2013
Abstract
We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional ∥u∥2Hs(Ω)+∫ΩW(u)dx, where ∥u∥Hs(Ω) denotes the total contribution from Ω in the H s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space Rn . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.
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