On the fractional Laplacian of variable order

Abstract: We present a novel definition of variable-order fractional Laplacian on Rn 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 function, for which we provide some instructive examples for specific problems.