Preprint

Inserted: 3 jul 2026
Last Updated: 4 jul 2026

Pages: 148
Year: 2026
Doi: 10.13140/RG.2.2.12383.62885
Links: https://doi.org/10.13140/RG.2.2.12383.62885

Abstract:

We prove capacitary screening results for periodically perforated global fractional Dirichlet energies. The holes are microscopic copies of a compact reference set, and the quantity governing the limit is their total fractional capacitary weight. This gives the full variational trichotomy: if this weight vanishes, the perforations are invisible in the $\Gamma$-limit; if it converges to a finite positive value, the limit energy acquires a zeroth-order capacitary reaction term; if it diverges and the reference perforation has positive fractional capacity, bounded-energy sequences collapse to zero in $L^2$. The use of the global Dirichlet form is essential. Exterior-tail interactions do not alter the critical one-hole capacitary constant, but they enter the compactness, lower-bound, recovery and coercive estimates and must be treated at the level of the full $H^s(\mathbb R^n)$ seminorm.

The same capacitary mechanism yields spatially inhomogeneous limits for modulated periodic arrays. When the microscopic size of each hole is scaled by a positive macroscopic factor, the critical screening density is multiplied by the corresponding $(n-2s)$-homogeneous factor, in accordance with the homogeneity of fractional capacity.

We also introduce, in a separated small-hole regime, a soft Robin-type capacitary model in which the hard constraint on the holes is replaced by a penalization carried by their fractional capacitary equilibrium measures. The effective one-hole response then interpolates between invisible obstacles and the hard-hole capacity. In particular, finite strengths produce an intermediate Robin-type capacitary reaction, while, at critical geometric density, the hard-hole capacitary coefficient is recovered in the singular limit of diverging soft strength.

Keywords: Homogenization, fractional Laplacian, nonlocal energies, perforated media, fractional capacity, capacitary screening

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