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

Abstract

We study existence, uniqueness and other geometric properties of the minimizers of the energy functionalwhere u H s (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 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.

Local behavior of fractional p-minimizers

“…For this, we will have to handle not only the usual nonlocal character of such fractional operators, but also the difficulties given by the corresponding nonlinear behavior. As a consequence, we can make use neither of the powerful framework provided by the Caffarelli-Silvestre s-harmonic extension ( [4]) nor of various tools as, e. g., the sharp 3-commutators estimates introduced in [5] to deduce the regularity of weak fractional harmonic maps, the strong barriers and density estimates in [26,28,29], the commutator and energy estimates in [25,27], and so on. Indeed, the aforementioned tools seem not to be trivially adaptable to a nonlinear framework; also, increasing difficulties are due to the non-Hilbertian structure of the involved fractional Sobolev spaces W s,p when p is different than 2.…”

Nonlocal phase transitions in homogeneous and periodic media

“…On the other hand, differently from the classical case, when s ∈ (0, 1) the existence of such solution u 0 is a rather delicate business, and it has been established, using variational and energy methods, in [PSV13], [CS15] and, in further generality, in [CP16].…”

Layered solutions for a fractional inhomogeneous Allen–Cahn equation

“…The previous heteroclinic connection w has been proved to exist and to be unique in [2,16]. We now describe our main results.…”