Local and global minimizers for a variational energy involving a fractional norm

We study existence, uniqueness and other geometric properties of the minimizers of the energy functional ‖u‖Hs(Ω) + ∫ Ω W (u) dx, where ‖u‖Hs(Ω) denotes the total contribution from Ω in the H 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 R. 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. 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 “ Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”. G. Palatucci Dipartimento di Matematica Dipartimento di Matematica Università degli Studi di Parma Università di Roma “Tor Vergata” Parco Area delle Scienze, 53/A Via della Ricerca Scientifica, 1 43124 Parma, Italia 00133 Roma, Italia E-mail: [email protected] O. Savin Department of Mathematics, Columbia University 2990 Broadway New York, NY 10027, USA E-mail: [email protected] E. Valdinoci Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma, Italia E-mail: [email protected] 2 Giampiero Palatucci et al.