Fractional Laplacians : A short survey

Abstract

<p style='text-indent:20px;'>This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).</p>