On the fractional Laplacian of variable order

Abstract

Abstract We present a novel definition of variable-order fractional Laplacian on $${\mathbb {R}}^n$$ R n based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poisson’s equation involving this operator and we compute the corresponding Green’s function, for which we provide some instructive examples for specific problems.