Sharp energy estimates for nonlinear fractional diffusion equations

Abstract

Abstract. We study the nonlinear fractional equation (−∆) s u = f (u) in R n , for all fractions 0 < s < 1 and all nonlinearities f . For every fractional power s ∈ (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension n = 3 whenever 1/2 ≤ s < 1. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equationIt remains open for n = 3 and s < 1/2, and also for n ≥ 4 and all s.

Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

“…On the other hand, energy estimates for minimizers of non-local energies have been independently obtained in [CC14] and [SV14] (in different settings). Since both these two results were set in a slightly different framework than ours, we provide their proofs in full details in Sections 2 and 3, respectively.…”

“…Estimate (3.2) has first been proved in [CC14] and [SV14] for the fractional Laplacian. While in the first paper the authors use the harmonic extension of u to R n+1 + to prove (3.2), in the latter work the result is obtained by explicitly computing the (5) We observe that, at this level, only the boundedness of W encoded in (W3) is relevant here.…”

Geometric inequalities for fractional Laplace operators and applications

“…For the fractional case, the analogue of this 1 − D symmetry result for stable solutions of the equation (−Δu) s = f (u) has been proven to be true in dimension 2 for any power 0 < s < 1 [4,23] and in dimension 3 for 1/2 ≤ s < 1 [2,3]. The proofs of all these results make use of the extension established in [5], which allows to study the fractional equation (1.1) by studying a local Neumann problem in the half-space R n+1 + .…”

“…The proofs of all these results make use of the extension established in [5], which allows to study the fractional equation (1.1) by studying a local Neumann problem in the half-space R n+1 + . In [23], the authors used a geometric inequality analogue to (1.11) for the extended problem in the half-space, while in [2][3][4] a different approach based on a Liouville type result is used.…”

Layered solutions for a fractional inhomogeneous Allen–Cahn equation

“…We will find a solution to (1) if we may construct a solution to (3). Denote w the unique solution of (−∂ xx ) s w − F (w(x)) = 0, w(0) = 0, w(±∞) = ±1.…”