On the Γ-limit of weighted fractional energies

Given p ∈ [1, ∞) and a bounded open set Ω ⊂ Rd with Lipschitz boundary, we study the Γ-convergence of the weighted fractional seminorm (Formula presented), as s → 1− for u ∈ Lp(Ω), where ˜u = u on Ω and ˜u = 0 on Rd \ Ω. Assuming that (fs)s∈(0,1) ⊂ L∞(Rd; [0, ∞)) and f ∈ Lipb(Rd; (0, ∞)) are such that fs → f in L∞(Rd) as s → 1−, we show that (1 − s)[u]ps,p,fs Γ-converges to the Dirichlet p-energy weighted by f 2. In the case p = 2, we also prove the convergence of the corresponding gradient flows.