Thin domains — regions where one dimension is much smaller than the others — arise in modeling membranes, films, and shells. Classical dimension reduction shows that PDE problems on thin 3D domains converge to 2D problems as the thickness approaches zero. The limiting 2D problem captures the dominant physics while discarding the thin-direction variation.
For fractional Sobolev seminorms — the nonlocal energy functionals that appear in fractional diffusion and Levy process models — the dimension reduction is more subtle. The fractional seminorm integrates a nonlocal kernel over pairs of points, and the kernel's interaction range competes with the domain thickness. When the range exceeds the thickness, the kernel “sees across” the thin dimension, and the reduction is nontrivial.
The authors establish the Gamma-convergence of fractional Sobolev seminorms on thin domains to a limit that depends on the fractional exponent. The limiting functional is still fractional, but on the lower-dimensional domain, with a modified kernel that accounts for the averaging over the thin direction.
The critical case is when the fractional exponent equals the reciprocal of the dimension ratio — the exponent at which the nonlocal interaction range matches the domain thickness. Above this exponent, the reduction is classical (the thin direction averages out). Below it, the nonlocal interactions dominate the thin-direction variation, and the limiting functional has a fundamentally different structure.
Dimension reduction for nonlocal energies. The thin direction doesn't just disappear — it gets absorbed into the kernel of the limit functional, modifying the nonlocal structure of the reduced problem.