Weighted nonlocal area functionals without the triangle inequality

TL;DR

This paper studies a nonlocal area functional with weights violating the triangle inequality, showing energy reduction via phase interface modification and proving convergence to a local functional as a parameter approaches one.

Contribution

It introduces a nonlocal energy model without the triangle inequality and demonstrates its convergence to a local area functional, expanding understanding of phase transition models.

Findings

Energy can be reduced by adding a phase 0 strip between phases -1 and 1.

Nonlocal energies $ ext{Gamma}$-converge to a local functional as $s o 1$.

The model remains lower semicontinuous despite lacking the triangle inequality.

Abstract

We consider a weighted nonlocal area functional in which the coefficients do not satisfy the triangle inequality. In the context of three phase transitions, this means that one of the weights is larger than the sum of the other two, say $$\sigma_{-1,1} > \sigma_{-1,0} + \sigma_{0,1}.$$ We show that the energy can be reduced by covering interfaces between phases $-1$ and $1$ with a thin strip of phase $0$. Moreover, as the fractional parameter $s\nearrow1$, we prove that the nonlocal energies $\Gamma$-converge to a local area functional with different weights. The functional structure of this long-range interaction model is conceptually different from its classical counterpart, since the functional remains lower semicontinuous, even in the absence of the triangle inequality.