2025/08/12 by Braides, Andrea, Solci, Margherita · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2508.08874
We consider the limit of squared Hs-Gagliardo seminorms on thin domains of the form Ωε=ω×(0,ε) in \mathbb Rd. When ε is fixed, multiplying by 1-s such seminorms have been proved to converge as s→ 1- to a dimensional constant cd times the Dirichlet integral on Ωε by Bourgain, Brezis and Mironescu. In its turn such Dirichlet integrals divided by ε converge as ε→ 0 to a dimensionally reduced Dirichlet integral on ω. We prove that if we let simultaneously ε→ 0 and s→ 1 then these squared seminorms still converge to the same dimensionally reduced limit when multiplied by (1-s) ε2s-3, independently of the relative converge speed of s and ε. This coefficient combines the geometrical scaling ε-1 and the fact that relevant interactions for the Hs-Gagliardo seminorms are those at scale ε. We also study the usual membrane scaling, obtained by multiplying by (1-s)ε-1, which highlighs the \em critical scaling 1-s∼|logε|-1, and the limit when ε→ 0 at fixed s.