Speaker:  Animesh Biswas, University of Nebraska-Lincoln

Title: Nonlocal Mean Curvature with Integral Kernel

Abstract: The focus of this talk will be on the recently introduced topic of nonlocal curvature. Several papers have studied nonlocal curvature realized via a kernel with nonintegrable singularity, which requires at least $C^{1,beta}$ regularity of the boundary. Nonlocal curvature of this form appears in many different applications, such as image processing, curvature driven motion, deformations. In this talk, we discuss the constant and ordered nonlocal mean curvature problem when the kernel is integrable. Constant nonlocal mean curvature problem with highly singular kernel was studied, independently by two separate groups: Ciraolo, Figalli, Maggi, Novaga, and respectively, Cabré, Fall, Solá-Moreles, Weth, where counterparts to Alexandrov’s theorem in the nonlocal framework were established. On the other hand, ordered mean curvature problem in the classical setting was introduced by Li and Nirenberg. Using Alexandrov's moving plane method we are able to solve that the solution to constant curvature problem is union of balls. On the other hand, in ordered curvature problem, we prove that solution to be symmetric around a hyperplane. Both of these works are in collaboration with Mikil Foss and Petronela Radu.