Sobolev-Poincaré implies John
Let \(\Omega\) be a domain with finite volume in \(\mathbb{R}^n\) and let \(1\leq p< n\). The domain \(\Omega\) satisfies the \(p\)-Sobolev-Poincaré inequality if there is \(C< \infty\) such that \(|u- u_\Omega|_{p^*}\leq C|\nabla u|_p\) for all smooth functions \(u\) in \(\Omega\). Here \(p^*= pn/(n- p)\) is the Sobolev conjugate of \(p\) and \(u_\Omega\) is the mean value of \(u\) over \(\Omega\). \textit{B. Bojarski} [Lect. Notes Math. 1351, 52-77 (1988; Zbl 0662.46037)] proved that John domains satisfy the Sobolev-Poincaré inequality for all \(1\leq p< n\); a domain is a John domain if it satisfies a twisted interior cone condition [the reviewer and \textit{J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I 4, 383-401 (1979; Zbl 0427.30021)]. Now, the authors prove the converse for simply connected domains in \(\mathbb{R}^2\) and a partial converse for general domains in \(\mathbb{R}^n\); an extra separation property for domains is needed in the general case.
scientific article; zbMATH DE number 858510