The Cheeger constant as limit of Sobolev-type constants (Q6594144)

Let \(\Omega\) be a bounded, smooth domain of \({\mathbb{R}}^N\) (\(N\ge2\)); for \(1<p<N\) and \(0<q<p^*=Np/(N-p)\), the quantity \N\[\N\lambda_{p,q}=\inf\Big\{\int_\Omega|\nabla u|^p\,dx\ :\ u\in W^{1,p}_0(\Omega),\ \int_\Omega|u|^q\,dx=1\Big\}\N\]\Nis considered, and the minimizer is denoted by \(u_{p,q}\). In [\textit{B. Kawohl} and \textit{V. Fridman}, Commentat. Math. Univ. Carol. 44, No. 4, 659--667 (2003; Zbl 1105.35029)], it is proved that \N\[\N\lim_{p\to1}\lambda_{p,p}=h(\Omega),\N\]\Nwhere \(h(\Omega)\) is the Cheeger constant \N\[\Nh(\Omega)=\inf\Big\{P(E)/|E|\ :\ E\subset\Omega,\ |E|>0\Big\},\N\]\Nbeing \(P(E)\) and \(|E|\) the perimeter of \(E\) and the Lebesgue measure of \(E\) respectively. The main result is that, when \(q(p)\) depends on \(p\), with \(\lim_{p\to1}q(p)=1\), then \N\[\N\lim_{p\to1}\lambda_{p,q(p)}=h(\Omega),\qquad\lim_{p\to1}\|u_{p,q(p)}\|_{L^1}=1,\qquad\lim_{p\to1}\|u_{p,q(p)}\|_{L^\infty}^{q(p)-p}=1.\N\]\NIn addition, for every sequence \(p_n\to1\) there exists a subsequence of \(u_{p_n,q(p_n)}\) converging in \(L^1\) to a nonnegative function \(u\in L^1\cap L^\infty\) such that \N\[\N\|u\|_{L^1}=1,\qquad1\le|\Omega|\|u\|_{L^\infty}\le\big(h(\Omega)/h(\Omega^*)\big)^N,\N\]\Nand for almost every \(t\ge0\) the \(t\)-superlevel set \N\[\NE_t=\big\{x\in\Omega\ :\ u(x)>t\big\}\N\]\Nis a Cheeger set.