On the Γ-limit of weighted fractional energies

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\R^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm % \begin{equation*} [u]_{s,p,f}^p = \int_{\R^d} \int_{\R^d} \frac{|\tilde u(x)- \tilde u(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\di x\di y \end{equation*} % as $s\to1^-$ for $u\in L^p(\Omega)$, where $\tilde u=u$ on $\Omega$ and $\tilde u=0$ on $\R^d\setminus\Omega$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\R^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\R^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\R^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}$ $\Gamma$-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.