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|Title:||Definition of fractional Laplacian for functions with polynomial growth|
|Authors:||Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico|
|Published in:||Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2318, ISSN 2198-5855|
|Publisher:||Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik|
|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 can 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 nonlinearity. 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.|
|Keywords:||Fractional operators; growth conditions; conditions at infinity|
|License:||This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.|
Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.
|Appears in Collections:||Mathematik|
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