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15 November 2019 On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program
Weiwei Ao, Hardy Chan, Azahara DelaTorre, Marco A. Fontelos, María del Mar González, Juncheng Wei
Duke Math. J. 168(17): 3297-3411 (15 November 2019). DOI: 10.1215/00127094-2019-0034

Abstract

We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case.

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Weiwei Ao. Hardy Chan. Azahara DelaTorre. Marco A. Fontelos. María del Mar González. Juncheng Wei. "On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program." Duke Math. J. 168 (17) 3297 - 3411, 15 November 2019. https://doi.org/10.1215/00127094-2019-0034

Information

Received: 22 March 2018; Revised: 23 April 2019; Published: 15 November 2019
First available in Project Euclid: 30 October 2019

zbMATH: 07154927
MathSciNet: MR4030366
Digital Object Identifier: 10.1215/00127094-2019-0034

Subjects:
Primary: 35J61
Secondary: 35R11, 53A30

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 17 • 15 November 2019
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