1、题目：Fully nonlinear nonlocal operators

   时间：7月15日15：00-16：30

   地点：格物楼503

   摘要：We consider equations involving fully nonlinear nonlocal operators $$ F_/alpha(u(x))/equiv/,C_{n,/alpha}PV/int_{/mathbb{R}^n}/frac{G(u(x)-u(z))}{|x-z|^{n+/alpha}}dz=f(x,u).$$

We prove a maximum principle and obtain key ingredients for carrying on the method of moving planes, such as narrow region principle and decay at infinity. Then we establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in a unit ball and in the whole space, as well as non-existence of solutions on a half space. We believe that the methods develop here can be applied to a variety of problems involving fully nonlinear nonlocal operators. We also investigate the limit of this operator as $/alpha/rightarrow2$ and show that $$ F_/alpha(u(x)) /rightarrow~a(-/Delta{u(x)}) + b|u(x)|^2.$$

2、题目：A priori estimates and existence of solutions for fractional equations

   时间：7月18日10：00-11：30

   地点：诚格物楼503

   摘要：We develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic  operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray-Schauder degree theory, we establish the existence of positive solutions.

We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.