On Poincaré-type integral inequalities
Let \(Q=[0,\sigma]^ n\). The author used very simple tools to prove for functions \(f,g\in C^ 1(Q)\) vanishing on \(\partial Q\) the estimates \[ \int_{Q}| f(x)g(x)| dx\leq \frac{\sigma^ 2}{8n}\int_{Q}[| \nabla f(x)|^ 2+| \nabla g(x)|^ 2]dx, \] \[ \int_{Q}[| f(x)| | \nabla g(x)| +| g(x)| | \nabla f(x)|]dx\leq \frac{\sigma}{2\sqrt{n}}\int_{Q}[| \nabla f(x)|^ 2+| \nabla g(x)|^ 2]dx. \] Here \(| \nabla f|^ 2=\sum^{n}_{i=1}| \partial f/\partial x_ i|^ 2\). The choice of \(g=f\) yields the Friedrichs-Poincaré inequality with a constant better then the ones usually mentioned in the literature.
multidimensional Poincaré-type integral inequalities
scientific article; zbMATH DE number 3966535