Wednesday, May 12, 2021 — 4:00 pm — online talk

Abstract: Our objective is to prove the existence of a smooth, embedded s-minimal (i.e. minimal with respect to the $s$-fractional perimeter) curve in a two dimensional compact manifold $M$. This result will be achieved through a suitable limit procedure of solutions of variational problems, originally due to Luciano Modica [1] for the classical perimeter. With a mountain-pass theorem we prove the existence of nontrivial critical points of the Allen-Cahn energy on the manifold.  Then, if we look at the interface of this critical point, that is the region of $M$ where this function jumps, performing a blow-down of the solution we show that this interface converges to a curve that is a local minimizer of the fractional perimeter.

References:

[1] Modica, L. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, 123–142 (1987). https://doi.org/10.1007/BF00251230