On the Γ-limit of weighted fractional energies

Given p∈[1,∞) and a bounded open set Ωᵈ with Lipschitz boundary, we study the Γ -convergence of the weighted fractional seminorm {equation*}[u]s,p,f^p=∫R^d ∫R^d {| u(x)- u(y)|^p}{\|x-y\|ᵈ⁺ˢᵖ}\,f(x)\,f(y)\,d x\,d y,{equation*} as s→1⁻ for u∈ Lᵖ(Ω) , where u=u on Ω and u=0 on RᵈΩ . Assuming that (fₛ)s∈(0,1)⊂ L^∞(Rᵈ;[0,∞)) and fb(Rᵈ;(0,∞)) are such that fₛ→ f in L^∞(Rᵈ) as s→1⁻ , we show that (1-s)[u]s,p,fₛᵖ Γ -converges to the Dirichlet p -energy weighted by f² . In the case $p=2$ , we also prove the convergence of the corresponding gradient flows.