Direct methods of moving planes, moving spheres, and blowing-ups for the fractional Laplacian

Direct methods of moving planes, moving spheres, and blowing-ups for the fractional Laplacian

-
Wenxiong Chen, Yeshiva University
Fine Hall 314

Many conventional approaches on partial differential operators do not work on the nonlocal fractional operator. To overcome this difficulty arising from non-localness, Caffarelli and Silvestre introduce the extension method to reduced the problem into a local one in one higher dimensions, which has become a powerful tool in studying such nonlocal problems and has yielded  a series of fruitful results. However, due to technical restrictions, sometimes one needs to impose extra conditions when studying the extended problems in higher dimensions, and these conditions may not be necessary if we investigate the original nonlocal problems directly. In this talk, we will introduce direct methods of *moving planes*, *moving spheres*, and *blowing-up and re-scaling arguments* for the fractional Laplacian.  By an elementary approach, we will first show the key ingredients needed in the {\em method of moving planes} either in a bounded domain or in the whole space, such as {\em strong maximum principles for anti-symmetric functions}, {\em narrow region principles}, and {\em decay at infinity}. Then, using simple examples, semi-linear equations involving the fractional Laplacian,  we will illustrate how this new {\em method of moving planes} can be conveniently employed to obtain symmetry and non-existence of positive solutions, under much weaker conditions than in the previous literatures. We firmly believe that these ideas and approaches can be effectively applied to a wide range of nonlinear problems involving fractional Laplacians or other nonlocal operators.