Motivated by generalizations of the Ginsburg-Landau energy and the diffusion equation in which derivatives are replaced by fractional derivatives, Caffarelli, Roquejoffre, and Savin studied the minimizers of a fractional perimeter functional on sets. Such minimizers have to satisfy a pointwise condition on their boundary, which can be used to define a notion of nonlocal mean-curvature. This definition holds for surfaces which are the boundary of a set. I will describe how to define a nonlocal notion of mean curvature for any surface by introducing a fractional area functional and considering its minimizers. This nonlocal mean-curvature can be used to motivate a nonlocal second fundamental form. I\u2019ll then go on to explain how the ideas in this definition can be used to define a fractional notion of length and an associated nonlocal curvature for a curve. Finally, I\u2019ll briefly explain how the same ideas can be used to define a fractional k-dimensional measure.<\/p>\n","protected":false},"excerpt":{"rendered":"
Motivated by generalizations of the Ginsburg-Landau energy and the diffusion equation in which derivatives are replaced by fractional derivatives, Caffarelli,…<\/p>\n