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Register a new userUniqueness and stability of the saddle-shaped solution to the fractional Allen-Cahn equation
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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.
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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-shaped solutions are doubly radial, odd with respect to the Simons cone ${(x, x) in mathbb{R}^m times mathbb{R}^m , : , |x| = |x|}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in narrow sets. The existence of the solution was already proved in part I of this work.
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 called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Delta)^s$ with $sin(0,1/2)$, as defined in Fourier space. In the singular limit $varepsilonto 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Omega$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^prime$-perimeter in $Omega$ for every $s^primein(0,1/2)$.
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 setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.
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 effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviors of solutions. More precisely, by deriving emph{partial} energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution $u(x,t)$ converges to a solution of an elliptic obstacle problem as $t to +infty$.
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