Definition of fractional Laplacian for functions with polynomial growth\tha

TL;DR

This paper extends the fractional Laplacian to functions with polynomial growth at infinity, providing a new definition and Schauder estimates useful for free boundary and blowup problems.

Contribution

It introduces a novel fractional Laplacian definition for functions with polynomial growth and establishes sharp Schauder estimates in this framework.

Findings

Defined fractional Laplacian for functions with polynomial growth.

Established sharp Schauder estimates controlling the solution's smoothness.

Framework applicable to nonlocal operators and free boundary problems.

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 H\"older 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.