On the $\mathrm{\Gamma }$-limit of weighted fractional energies

Given $p\in [1,\mathrm{\infty })$ and a bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with Lipschitz boundary, we study the $\mathrm{\Gamma }$-convergence of the weighted fractional seminorm $$[u{]}_{s,p,f}^p={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\frac{\mathrm{\mid }\tilde{u}(x)-\tilde{u}(y){\mathrm{\mid }}^p}{\mathrm{\parallel }x-y{\mathrm{\parallel }}^{d+sp}}f(x)f(y)\mathrm{d}x\mathrm{d}y$$ as $s\to 1^{-}$ for $u\in L^p(\mathrm{\Omega })$, where $\tilde{u}=u$ on $\mathrm{\Omega }$ and $\tilde{u}=0$ on ${\mathbb{R}}^d\setminus \mathrm{\Omega }$. Assuming that $(f_s{)}_{s\in (0,1)}\subset L^{\mathrm{\infty }}({\mathbb{R}}^d;[0,\mathrm{\infty }))$ and $f\in {\mathrm{L}\mathrm{i}\mathrm{p}}_b({\mathbb{R}}^d;(0,\mathrm{\infty }))$ are such that $f_s\to f$ in $L^{\mathrm{\infty }}({\mathbb{R}}^d)$ as $s\to 1^{-}$, we show that $(1-s)[u{]}_{s,p,f_s}$ $\mathrm{\Gamma }$-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.