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In this paper we prove the uniqueness of the saddle-shaped solution to the semilinear nonlocal elliptic equation $(-Delta)^gamma u = f(u)$ in $mathbb R^{2m}$, where $gamma in (0,1)$ and $f$ is of Allen-Cahn type. Moreover, we prove that this solution is stable whenever $2mgeq 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone ${(x, x) in mathbb R^{m}times mathbb R^m : |x| = |x|}$ is a stable nonlocal $(2gamma)$-minimal surface in dimensions $2mgeq 14$. Saddle-shaped solutions of the fractional Allen-Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions $2$, $4$ and $6$. Thus, after our result, the stability remains an open problem only in dimensions $8$, $10$, and $12$. The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.
This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-sha
We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Omegasubsetmathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly calle
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Omega$). After sett
This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness, smoothing effe