Poincaré inequalities and Steiner symmetrization (Q1815228)

The domain \(\Omega\subset\mathbb{R}^n\) is said to be a \(p\)-Poincaré domain, \(1\leq p<\infty\), provided it supports the \(p\)-Poincaré inequality \[ \int_\Omega|u- u_\Omega|^p dx\leq M\int_\Omega|\nabla u|^p dx,\quad u\in W^1_p(\Omega), \] where we denoted \(u_\Omega= \int_\Omega u(x)dx\). The geometry of Poincaré domains is quite complicated and a complete geometric characterization remains an elusive unsolved problem. In this paper, the authors give a geometric characterization of \(p\)-Poincaré domains, restricted to the class of Steiner symmetric domains \(\Omega\subset\mathbb{R}^n\). The characterization works only for \(p>n-1\). The authors give also a more restricted class of Steiner symmetric domains for which the characterization remains valid for all \(p>1\).