Fractional measure and nonlocal curvature: surfaces, curves, and beyond – Brian Seguin (Loyola University Chicago)


Aula Seminari


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’ll 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’ll briefly explain how the same ideas can be used to define a fractional k-dimensional measure.

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