Weighted nonlocal area functionals without 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.