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We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form $$ (-\Delta)^{s/2} u=u-u^3,$$ with~$s\in(0,1)$. More generally, we can take into account equations like $$ L u = f(u),$$ where $f$ is a bistable nonlinearity and $L$ is an integro-differential operator, possibly of anisotropic type. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in the companion paper.

We study the viability of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances—namely, a precise condition on the distribution of the resource—under which the introduction of a new nonlocal dispersal behavior is favored with respect to the local dispersal behavior of the resident population. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can invade an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribution of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic environmental factors that favor nonlocal strategies.

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric., Comment: 3 figures. Minor changes with respect to the previous version. Accepted for publication on Journal d'Analyse Math\'ematique

We consider here solutions of the nonlinear fractional Schrodinger equation We show that concentration points must be critical points for V. We also prove that if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such a minimal point as e tends to zero. In addition, if the potential V is radial and radially decreasing, then the minimizer is unique provided e is small.

We study bounded, monotone solutions of$\Delta u=W'(u)$ in the whole of$\R^n$, where$W$ is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, $u$ is $1$D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of $\Gamma$-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are $1$D, at least up to dimension 4.

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely ∫ Ω | ∇ u ( x ) | 2 d x + Per σ ( { u > 0 } , Ω ) , with σ ∈ ( 0 , 1 ) . We obtain regularity results for the minimizers and for their free boundaries ∂ { u > 0 } using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.