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.

On higher-dimensional singularities for the fractional yamabe problem: A nonlocal mazzeo-pacard program / Ao, W.; Chan, H.; Delatorre, A.; Fontelos, M. A.; Del Mar Gonzalez, M.; Wei, J.. - In: DUKE MATHEMATICAL JOURNAL. - ISSN 0012-7094. - 168:17(2019), pp. 3297-3411. [10.1215/00127094-2019-0034]

On higher-dimensional singularities for the fractional yamabe problem: A nonlocal mazzeo-pacard program

Delatorre A.;Wei J.
2019

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.
2019
Semilinear elliptic equations; fractional partial differential equations; conformal differential geometry
01 Pubblicazione su rivista::01a Articolo in rivista
On higher-dimensional singularities for the fractional yamabe problem: A nonlocal mazzeo-pacard program / Ao, W.; Chan, H.; Delatorre, A.; Fontelos, M. A.; Del Mar Gonzalez, M.; Wei, J.. - In: DUKE MATHEMATICAL JOURNAL. - ISSN 0012-7094. - 168:17(2019), pp. 3297-3411. [10.1215/00127094-2019-0034]
File allegati a questo prodotto
File Dimensione Formato  
Ao_On-higher-dimensional_2019.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 657.99 kB
Formato Adobe PDF
657.99 kB Adobe PDF   Contatta l'autore

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1590795
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 28
  • ???jsp.display-item.citation.isi??? 24
social impact