Higher-order fractional Laplacians: An overview

Abstract

We summarize some of the most recent results regarding the theory of higher-order fractional Laplacians, i.e., the operators obtained by considering (non-integer) powers greater than 1 of the Laplace operator. These can also be viewed as the nonlocal counterparts of polylaplacians. In this context, nonlocality meets polyharmonicity and together they pose new challenges, producing at the same time surprising and complex structures. As our aim is to give a fairly general idea of the state of the art, we try to keep the presentation concise and reader friendly, by carefully avoiding technical complications and by pointing out the relevant references. Hopefully this contribution will serve as a useful introduction to this fascinating topic.

Keywords

• maximum principle
• integration by parts formulas
• unique continuation
• representation formulas
• positivity preserving property