Г-convergence of Oscillating Thin Obstacles

Eurasian Mathematical Journal (Казахстан). 2013. Vol. 4. No. 4. P. 88–100.

Koroleva Y., Stromqvist M.

 

We consider the minimization problems of obstacle type

min{∫Ω|Du|2dx:u≥ψε on P, u=0 on ∂Ω},

as ε→0. Here Ω is a bounded domain in Rn, ψε is a periodic function of period ε, constructed from a fixed function ψ, and P⊂⊂Ω is a subset of the hyper-plane {x∈Rn:x⋅η=0}. We assume that n≥3 and that the normal η satisfies a generic condition that guarantees certain ergodic properties of the quantity

#{k∈Zn:P∩{x:|x−εk|<εn/(n−1)}}.

Under these hypotheses we compute explicitly the limit functional of the obstacle problem above, which is of the type

H10(Ω)∋u↦∫Ω|Du|2dx+∫PG(u)dσ.

Keywords: homogenization theory, obstacle problem, теория усреднения