Uniqueness for linear integro-differential equations in the real line and applications

Abstract

In this work we prove the uniqueness of solutions to the nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ L φ - c ( x ) φ = 0 in $$\mathbb {R}$$ R , where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $$L u = f(u)$$ L u = f ( u ) in $$\mathbb {R}$$ R when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

The stickiness property for antisymmetric nonlocal minimal graphs

“…Some comments about antisymmetric functions. Maximum principles in the presence of an odd symmetry have been already treated in the field of integrodifferential equations (see for instance [38,33,19,32,30]). The main idea in this setting is that the operator can be rewritten as a different integro-differential operator acting only on functions defined in the halfspace, but still with a positive kernel (under certain assumptions on the original operator).…”

“…This and (25) lead to (32), as desired. By (32), we know that the supersolution condition (24) holds true for P := (p, u(p)).…”