Abstract
We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional
where \({\|u\|_{H^s(\Omega)}}\) 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 \({\mathbb{R}^n}\) . 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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The first author has been supported by Istituto Nazionale di Alta Matematica “F. Severi” (Indam) and by ERC grant 207573 “Vectorial problems”. The second author has been partially supported by National Science Foundation (NSF) grant 0701037. The third author has been partially supported by FIRB “Project Analysis and Beyond”. The second and the third authors have been supported by ERC grant 277749 “\({\epsilon}\) Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”.
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Palatucci, G., Savin, O. & Valdinoci, E. Local and global minimizers for a variational energy involving a fractional norm. Annali di Matematica 192, 673–718 (2013). https://doi.org/10.1007/s10231-011-0243-9
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DOI: https://doi.org/10.1007/s10231-011-0243-9