On the Γ-limit of weighted fractional energies

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