Abstract

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

Keywords: Poincaré-Friedrich's inequality, $L^p$-space.

MSC: 26D10, 35P15

Pages:  11$-$14     

Volume  57 ,  Issue  1$-$2 ,  2005