Preprint

Inserted: 9 oct 2026

Year: 2026
Doi: https://doi.org/10.13140/RG.2.2.34009.89445

Abstract:

We investigate $s$-perimeter functionals with prescribed exterior data in the singular limit $s \to 0^+$. In this regime, although the leading-order asymptotics depends on the global distribution of the exterior datum, it may collapse to a purely volumetric functional and therefore fail to distinguish among different competitors. To capture the first nontrivial variational contribution beyond this degenerate scenario, we establish a full $\Gamma$-expansion; the resulting limit functional consists of a spatial interaction term measuring the residual interaction between the phase inside the domain, and a scalar far-field correction accounting for the prescribed exterior datum.

Applications to unconstrained and volume-constrained minimization problems show that perturbations, invisible at the zero-order, may nevertheless select the limiting shape of minimizers. This provides a selection principle for small-order fractional perimeter problems and makes explicit how the prescribed exterior datum affects the variational problem as $s\to0^+$.