Definition of fractional Laplacian for functions with polynomial growth

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition, which (in addition to the various results that we prove here) can also be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the forcing term. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.

Original languageEnglish
Pages (from-to)1079-1122
Number of pages44
JournalRevista Matematica Iberoamericana
Volume35
Issue number4
DOIs
Publication statusPublished - 1 Jan 2019

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Fractional Laplacian
Polynomial Growth
Schauder Estimates
Infinity
Seminorm
Forcing Term
Free Boundary Problem
Operator
Equivalence class
Blow-up
Modulo
Linearly
Norm
Polynomial

Cite this

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abstract = "We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition, which (in addition to the various results that we prove here) can also be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth H{\"o}lder norm of the solution is controlled in terms of the seminorm of the forcing term. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.",
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Definition of fractional Laplacian for functions with polynomial growth. / Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico.

In: Revista Matematica Iberoamericana, Vol. 35, No. 4, 01.01.2019, p. 1079-1122.

Research output: Contribution to journalArticle

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AU - Valdinoci, Enrico

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