On the $\Gamma$-limit of weighted fractional energies

Abstract

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 \[ [u]_{s,p,f}^p = \int_{\mathbb R^d} \int_{\mathbb R^d} \frac{|\tilde{u}(x)- \tilde{u}(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y \] as $s\to 1^{-}$ 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 $s\to 1^{-}$, 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.

18 pages

References in corpus (5)

• Gamma-convergence of nonlocal perimeter functionals
• Asymptotics of weighted Gagliardo seminorms
• Second-order asymptotics of fractional Gagliardo seminorms as $s\to 1^{-}$ and convergence of the associated gradient flows
• Sharp conditions for the BBM formula and asymptotics of heat content-type energies
• Gamma-convergence as $s\to 1^{-}$ of anisotropic nonlocal fractional perimeter functionals