Harnack's inequality for fractional nonlocal equations

Pablo Raúl Stinga,Chao Zhang

doi:10.3934/dcds.2013.33.3153

Abstract

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operatorsarising in classical orthogonal expansions and the radial Laplacian.To get the results we use an analytic method based on ageneralization of the Caffarelli--Silvestre extension problem, theHarnack's inequality for degenerate Schrödinger operators provedby C. E. Gutiérrez, and a transference method. In this manner weapply local PDE techniques to nonlocal operators. Onthe way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.

Highlights

Very recently, a great deal of attention was given to nonlinear problems involving fractional integrodifferential operators
A great deal of attention was given to nonlinear problems involving fractional integrodifferential operators. These problems arise in Physics and Mathematical Finance, among many other fields, see for instance [5, 6, 8, 15, 21, 22] and the references therein
One of the tools that plays a crucial role in the regularity theory of PDEs is Harnack’s inequality, see for example [6, 7, 9, 10, 11, 12, 19, 23, 25, 28]

Summary

Introduction

A great deal of attention was given to nonlinear problems involving fractional integrodifferential operators. In this paper we show interior Harnack’s inequalities for solutions of nonlocal equations given by fractional powers of second order partial differential operators. Fractional operator, Harnack’s inequality, degenerate elliptic equation, Schrodinger operator, heat-diffusion semigroup, Liouville theorem, maximum and comparison principle. First we use two tools: the extension problem of [23] and Harnack’s inequality for degenerate Schrodinger operators of C. As a by-product of our method, we obtain a Liouville theorem for fractional powers of divergence form elliptic operators on Rn, see Remark 3.3.

Harnack’s inequality for fractional Schrodinger operators

Classical orthogonal expansions

Laplacian and Bessel operators