Fractional Laplacian in conformal geometry

By S. Chang, María del Mar González. 2010

444 citations recorded by the source corpus.

Abstract In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli and Silvestre and a class of conformally covariant operators in conformal geometry.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b38d8f9b2cdfe790581773d0ad22aae32481a7d4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On finite Morse index solutions of higher order fractional Lane-Emden equations (2014, unknown): “See also Case-Chang [3] and Chang-Gonzales [4].”
Singular Solutions of Fractional Order Conformal Laplacians (2009, result): “One other way that ∆γ arises is as a generalized Dirichlet to Neumann map; this definition is essentially the same as the one above involving the scattering operator (indeed, the point of view in geometric scattering theory is that the scattering operator is simply the Dirichlet to Neumann operator ”
On Fractional GJMS Operators (2014, methodology): “1 relies only on conformal covariance, and so one could simply compute as in [6] to establish a more general form of Theorem 1.”
A fractional Yamabe flow and some applications (2011, methodology): “Recently, Chang and González [21] reconciled the way of Graham and Zworski to define conformally invariant operatorsP gσ of non-integer order σ ∈ (0, n2 ) and the localization method of Caffarelli and Silvestre [18] for factional Laplacian (−∆)σ on the Euclidean spaceRn.”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, background): “See also [14, 22] and [34], where this observation appears in the contexts of conformal geometry and stochastic processes, respectively.”
Extension problem and fractional operators: semigroups and wave equations (2012, methodology): “2) given in [6] has been used to show that the fractional Laplacian (−∆)σ coincides with a certain conformally covariant operator Pσ on the hyperbolic space R n+1 + from which R n is to be seen as its boundary [9].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, methodology): “Such observation in [11] plays a fundamental role in this paper and provides an alternative way to study the fractional partial differential equation (2.”
On higher order extensions for the fractional Laplacian (2013, background): “If we take the point of view of considering the extension problem as a version of the scattering problem a la [2], we have recovered our energy, ∫ (∆U)2dxdy, as the equivalent of the finite part, of the energy integral ∫ y−2|∇U |2 (see, e.”
Some constructions for the fractional Laplacian on noncompact manifolds (2012, methodology): “In [16] both the geometrical and the PDE points of view were reconciled and, in particular, the fractional Laplacian on the sphere S (or R through stereographic projection) is defined from scattering theory in the Poincaré ball.”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, methodology): “The fractional analogue of the Yamabe problem has been introduced in [19] and was also studied in detail, see e.”
Indefinite fractional elliptic problem and Liouville theorems (2014, background): “lous dynamics, ﬂames propagation and chemical reactions in liquids, population dynamics, geophysical ﬂuid dynamics, and American options in ﬁnance. It also has connections to conformal geometry, e.g. [CG]. 1991 Mathematics Subject Classiﬁcation. 35B53, 35B45, 35J70, Key words and phrases. fractional”
Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions (2019, methodology): “Another point of view that relates the class of operators we consider with previous works is the one related to scattering theory and in particular the link between the extension property exploited by Caffarelli and Silvestre and scattering theory as in [11] as pointed out (and further explored) by ”
Fractional Laplacian on the torus (2012, methodology): “3) was obtained by Chang and González in [8].”
SHARP HARDY-LITTLEWOOD-SOBOLEV INEQUALITY ON THE UPPER HALF SPACE (2013, result): “Our results are closely related to the study of fractional Lapalacian operator in the whole space [8], fractional Yamabe problem [9, 22, 23, 25, 33, 34], fractional 6”
Direct Method of Moving Spheres on Fractional Order Equations (2015, background): “wed as the conformal inﬁnity of (Hn+1,gH). Speciﬁcally, the conformal compactiﬁcation is given by the following coordinates, gH = dt2 +(1− t2 4 )2g 1 t2 ,t∈ [0,2]. (4.1) Clearly, (t2g H)|t=0 = g1. By [ChGon], for any representative h ∈ [g1], the fractional GJMS operator Pα[h] ≡ Pα[h,gH] can be deﬁne”