Inequality of Poincaré-Friedrich's type on $L^p$ spaces

Abstract

In this paper it is demonstrated that the inequality$$\biggl(ınt_G|u|^p\,dx\biggr)^{1/p}\leq A_p\biggl(ınt_D|\nabla u|^p\,dx\biggr)^{1/p},\quad u|_{\partial D}=0,1\leq p\leqınfty$$holds, where $G\subset D\subset R^2$, $D$ is a convex domain and constant $A_p$ isexpressed in terms of areas of $G$ and~$D$.