Direct methods on fractional equations

Abstract

In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

Monotonicity of solutions for a class of nonlocal Monge-Ampère problem

“…However, the application of this two methods needs extra conditions to be imposed on the solutions. Moreover, they do not work for nonlinear nonlocal operators, such as the fractional p-Laplacian(see [13] for details). Thanks to the works of Chen and Li et al in [9,10,11], direct methods of moving planes are introduced for the fractional Laplacian and fractional p-Laplacian without going through extensions or integral equations, which have been applied to obtain symmetry, monotonicity, and non-existence of solutions for various semi-linear equations involving these nonlocal operators.…”

Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

“…If assumptions (A1)-(A4) and (H4) hold, then system (1) has unique weak solution u for any ∈ . We consider the control problem as follows:…”

“…A natural question arises: Whether we can extend the results of Laplacian problems to the fractional Laplacian ones or not? Unfortunately, these extensions are not always true, see [1][2][3]. In particular, Devillanova and Carlo Marano have studied the following fractional differential equation in [3] Devillanova and Carlo Marano have delicately compared the nonfractional cases ( = β 0, 1) and the fractional case ( < < β 0 1) through the mathematical analysis and experimental data.…”

Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators

“…However, it seems that the extension method and the integral equations method don't work for more general nonlocal operators like A α , the fractional p-Laplacian, etc., see [10,16,23]. Recently, Chen-Li-Li [13] systemically developed a so-called direct moving plane method which can handle (−∆) α 2 directly.…”