Abstract: There is a triple point of geometry, mechanics and mathematical analysis where three notions are met: the geometrical notion of mean curvature of a surface plays a pivotal role between the mechanical notion of diusion of a substance and the analytical notion of Laplacian of a scalar-valued function. Interestingly, this triple-point situation occurs also for nonlocal mean curvature, anomalous diusion, and fractional Laplacians [1]. So far, only closed surfaces have been considered, for which nonlocal directional and mean curvatures have been dened [2, 3]. I will discuss the analogous notions for open surfaces recently proposed in [4], which are based on a notion of area functional for this type of surfaces.

References:

[1] P. Podio-Guidugli, A notion of nonlocal Gaussian curvature. Rend. Lincei Mat. Appl. 27 (2016), 181-193.

[2] N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature. Numer. Funct. Anal. Optim. 35 (2014), 793-815.

[3] X. Cabre, M.M. Fall, J. Sola-Morales, and T. Weth, Curves and surface with constant nonlocal mean curvature: meeting Alexandrov and Delauney. arXiv:1503.00469 [math.AP].

[4] R. Paroni, P. Podio-Guidugli, and B. Seguin, On the nonlocal curvatures of open surfaces. arXiv:1701.06513 [math.DG].