The Cheeger constant as limit of Sobolev-type constants

Abstract

Let $\Omega$ be a bounded, smooth domain of ${\mathbb{R}}^N,$ $N\ge 2.$ For $1<p<N$ and $0<q(p)<p^{\ast }:=\frac{Np}{N-p}$ let \[ λ_{p,q(p)}:=\inf\left\{ \int_Ω\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(Ω)\text{ \ and \ }\int_{Ω}\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} . \] We prove that if ${\lim }_{p\to 1^{+}}q(p)=1,$ then ${\lim }_{p\to 1^{+}}{\lambda }_{p,q(p)}=h(\Omega )$, where $h(\Omega )$ denotes the Cheeger constant of $\Omega .$ Moreover, we study the behavior of the positive solutions $w_{p,q(p)}$ to the Lane-Emden equation $-\mathrm{div}({\left|\nabla w\right|}^{p-2}\nabla w)={\left|w\right|}^{q-2}w,$ as $p\to 1^{+}.$

16 pages. Typing errors have been corrected, one reference has been added, and the abstract has been slightly modified

References in corpus (2)

• Solutions of the Cheeger problem via torsion functions
• Absolute continuity of the best Sobolev constant of a bounded domain