Geometric inequalities for fractional Laplace operators and applications

Abstract

We prove a weighted fractional inequality involving the solution u of a nonlocal semilinear problem in R n . Such inequality bounds a weighted L 2 -norm of a compactly supported function φ by a weighted H s -norm of φ. In this inequality a geometric quantity related to the level sets of u will appear. As a consequence we derive some relations between the stability of u and the validity of fractional Hardy inequalities.Mathematics Subject Classification. Primary 35J22; Secondary 35B65.

Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain

“…The non-analyticity of the correlators (14), and especially (16), in general overlooked in the literature, is required in order to achieve C 12 → 0 for large time-or space-separations, viz. All two-point functions (12,14,16) have indeed the symmetries C 12 (t 1 , t 2 ; r, r) = C 21 (t 2 , t 1 ; r, r) and C 12 (t, t; r 1 , r 2 ) = C 21 (t, t; r 2 , r 1 ), under permutation ϕ 1 ↔ ϕ 2 of the two scaling operators, as physically required for a correlator. The shape of the scaling functions of these three two-point functions is compared in figure 3.…”

“…We had already mentioned in section 1 (Comment 1), that if we restrict to case A and take x = x 1 = x 2 = 1 2 and ν > 0, the resulting two-time response F (t, r) = F 0 t 1−2x ε 1 νt/(ν 2 t 2 + r 2 ), with t = t 1 − t 2 and r = r 1 − r 2 , reproduces the exact solution (6b). We stress that no choice of x 1 will make the ortho-conformal prediction (12) compatible with (6b). This is the main conceptual point of this work: the non-local representation (29) of the meta-conformal algebra ev is necessary to reproduce the correct scaling behaviour of the nonstationary response of the dle process.…”

“…The form of the meta-1 conformal two-point function (14), is clearly different for finite values of the scaling variable v = (r 1 − r 2 )/(t 1 − t 2 ), and similarly for the galilean conformal case (16). The ortho-conformal two-point function (12) looks to be much closer, with the choice x 1 = 1 2 and the scale factor fixed to ν = 1, were it not for the extra factor ν(t − s).…”

“…Formulating (1) in direct space requires the Riesz-Feller fractional derivative [64,53,12] of order α.…”

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

“…Here is how this article is structured. In Section 2, we establish a Poincaré type inequality for stable solutions of (1.5) with a general kernel K. This inequality is inspired by the ones given originally by Sternberg and Zumbrun in [31,32] and later in [17,[21][22][23]. In Section 3, we apply the Poincaré inequality to establish our main result that is one-dimensional symmetry of solutions for (1.5) in two dimensions when the kernel is either with finite range or with decay at infinity.…”